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 functionÕs output can be another functionÕs
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 34. 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 studentsÕ 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, Òthe 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 Òlesson logicÓ to your colleagues
to describe your general approach (and the rationale for your approach) to
teaching function composition conceptually.Ó 
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: sˆu), 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)). 


