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.

Step	Action	Reason
	Show a simulation of a growing circle and ask students to determine what is changing in the situation. Ask students to identify pairs of quantities (variables) that are changing in the situation. Note: The original simulation will be illustrated in a computer environment—the flash program has been developed for the growing circle simulation.	To get students to form a dynamic image of the growing circle and to become comfortable in recognizing quantities (variables) that are changing in the situation and to attend to how pairs of quantities are changing in tandem.
	Ask different students in the class to describe the quantity (variable) pairs that they chose and to describe how the variables change together. (Have all students in the class use the finger tool to coordinate how each pair of variables changes in tandem.) Note: Some pairs that will likely be mentioned include: (radius, circumference); (radius, area); (circumference, area); (time, radius). It is important that the class consider (radius, area) and (time, radius)	This gives students experience in identifying the quantities (variables) that change in a dynamically changing situation. Asking students to use the finger tool to enact how variable pairs change in tandem gives them experience imagining the variables changing in tandem and representing how the variables change together.
	Have students respond to the following problem in groups of 3-4. Problem: An oil spill causes a a film of oil to radiate outward from a ship. The Clean Environment Company needs to develop a way of finding the area formed by concentric circles that travel outward at various times after the oil begins leaking from the ship. The captain of the ship observed that the radius of the concentric circles is growing at a rate of about 3 kilometers each minute for the first 7 minutes after the spill. Draw a picture of the situation and imagine what is changing.	Sets up a problematic situation that requires students to define a new function by composing two functions. Motivates the need for developing and defining a new function by composing two other functions. Lays the groundwork for thinking about what two functions need to be composed to write a function that defines area in terms of time.
	Ask students to explain how radius; then area varies as time changes from 0 to 1 minute; from 1 to 2 minutes; from 2 to 3 minutes, etc.	This will provide students’ practice in using covariational reasoning to imagine how pairs of quantities change in tandem. It should also motivate students to think about what two processes need to be composed to arrive at a function to define area in terms of time.
	Now, ask students to create, using graphing calculator, graphs of two functions: one that expresses radius as a function of time; and the second that expresses area as a function of radius. Ask students to: i) create a table of values that determine, first radius, then area when time = 1, 2, 3 and 4. In what order would you evaluate these functions to find the area when you know the time? Use these two graphs to find: i) the area when the radius is 1; ii) the area when the radius is 2; iii) the area when the radius is 3. Describe the order in which the functions are evaluated. Explain what A(r(2)) means using the graph and the table. If you know the area, can you describe a way to find the time that has elapsed since the oil spilled, assuming the concentric circles continue traveling outward at the same rate? (Review the notion of inverse function before they start—use the table, graph and formula that defines radius as a function of time to create the inverse function that defines time as a function of radius (i.e., write a function that will determine the value of time, when given values for the radius t(r) = r/3). Now, write a function that will give the value of the radius, when given the area. What two functions must be chained together to determine the time that has elapsed for various values of the area?	This should provide students practice in stringing two function processes together to create one new process; one that relates the input of one process with the output of the second.
	Ask students to describe another situation (in a science context) that requires the use of function composition. Have students discuss their examples in their groups, and pick one context to build a general model and explore in all representational contexts as they did in item 6 above.	Should help teachers recognize the usefulness of function composition. Should also help teachers recognize situations that can be modeled by chaining two processes together.
	Tell the students (teachers) that, “the other teachers that work at your school have indicated that they do not understand how to teach the idea of function composition conceptually. Write a letter or “lesson logic” to your colleagues to describe your general approach (and the rationale for your approach) to teaching function composition conceptually.”	Should help students (teachers) move their thinking along as to how they would teach function composition.
	Recap what they have done: 1) Understand what is involved in composing two functions. 2) Recognize that it is sometimes useful to be able to create a new function by defining it in terms of two other functions. 3) Recognize that chaining two functions in sequence is possible when the output variable of one is the same as the input variable of the other. As an example, composing two functions is useful when you want to define a function h that maps s to u (i.e., h: sàu), but you only know a function, call it f that maps t to u. This sets up a need to find a new function g that maps s to t. This will allow you to define h by composing f and g, written as g(f(s)).