Compositions in Context

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SWBAT compose functions and interpret the meaning of the compositions in context.

Big Idea

Students first become quick function composers with a relay game and then work to see how composite functions can be used to model cell phone bills.

Warm-up: Relay Game

15 minutes

Preparation: Prepare one set of relay cards containing each of the functions for each group. Print the sheets of functions on different colors of cardstock and laminate each sheet; then cut the functions out and put one of each color in a sandwich bag for easy distribution and collection.

Narrative: Divide students in groups of four or five. In groups of five, each person will receive a card containing a different function. For groups of four, one student in each group will receive two cards; be sure to have the groups rotate who has two cards from round to round.

Each group should have one sheet of paper on which to work the problems. To begin the relay, the teacher calls out a number. The student with function number one evaluates his function for the number called by the teacher; he must show work (at least substitution of the number into the function) and write the answer to his step on the group’s paper. After evaluating the function for the number called out, that student passes the group’s answer sheet to the person with function number two; that student then substitutes the value obtained by the first student into function number two. After evaluating function number two using the answer obtained by the first student, the paper is then passed to the student with function number three; that student evaluates function number three using the value obtained by the second student. This continues until all five functions have been evaluated using the value obtained by the previous student.

At the conclusion of round one of the relay, each group should have work and an answer for each step of the relay. As groups finish, they show their work to the teacher who will tell them if their answer is correct or not. If it is not, the group may work together to fix their answer sheet; after finding their error(s), they may resubmit their answer. I also require students to write a complete statement for this answer. I want them to have the correct function notation on one side of the equation. For example: 
Teachers may use a variety of methods to keep score, or they may choose to reward the winning group after each round. I usually award points for each round based on the order in which groups finish. At the end of the activity, the winning group receives some type of reward; of course, with some classes, bragging rights are the only motivation needed. After round one, have students switch cards so that they will have practice with different types of functions. Proceed with round two by calling out a new number to be used to evaluate the first function with.

Play for as many rounds as you have time. I am guessing I will probably only do two round of it before moving on to the next activity. I also will allow students to use calculators. It is great practice for them to use the calculator as we should hear the teams working together and really talking about the correct way to enter these expressions accurately in order to get the correct answer.


It is important that students take away from this activity that a function is simply a rule that takes an input and returns a single output. If you have two functions, then you can obtain a new function by the rule:

1. start with an input

2. obtain an output from the first function

3. use that output as the input for the second function to obtain a final output.

This is an important concept to understand and will really help students in completing a table of values for compositions (like problem 2 on the homework).

Discussion: Mathematical Practice of the Day

5 minutes

Today, I would like student to think more deeply about Mathematical Practice 4: Model with Mathematics. To help students better understand this practice I am going to present page 2 in the flipchart. I want to talk with students about the use of multiple representations in math to model a situation. A good mathematician is fluent in all of these representations and can easily go back and forth between them.

In class, we will be practicing how to go from a verbal representation to an algebraic representation. I ask the class whether or not they feel we have practiced using multiple representations already. Hopefully students recognize that we have been looking at equations for functions and making tables and graphs based on those equations. Students have also worked with tables of values and graphs. I think it is difficult to work from graphs and tables and generate equations. So, that is our focus for today!

We have written equations in several earlier lessons: shifting graphs, finding the rules for function machines, writing the equation of piecewise functions, etc. Today, students practice modeling descriptions of problems using algebraic equations. We all know is a difficult task for students! (Well, at least for mine!)

Group Work: Composite Functions in Context

20 minutes

Next, students will complete the compositions in context problems as a team. These problems should help students to see how composition of functions can be used in everyday life. Also, I think with the context of a real world situation it will help students to develop more of an understanding of what it means to compose two functions. Students should present their findings at the end of class if there is time.

While working in their teams, I am going to encourage students to practice TTTT - Team, Team, then Teacher. For more information on the first problem and how I plan to help make this text messaging problem accessible to all check out the video narrative below from 0:36-2:34. For the same information on the second problem, the discount problem, check out the video from 2:35-3:47. The last part of the video addresses student presentations.

Closure: Group Presentations

10 minutes

With about 10 minutes remaining in class, have selected teams present their findings. I plan to have each team only talk about one section, but have at least 4 different teams present. Teams will have 2 minutes to present so I know that is a small allotment of time. If conversations are robust don’t cut them short, instead just cover the first two bulleted items. If students really understand this first problem they should be able to make the connections in the second problem.

Here are some prompts I plan to ask students to talk about while presenting to really draw out the key concepts I wanted covered today:

  • Explain your thought process, as you would to somebody in a freshman Algebra class, for how you found the functions for t(f) and g(t).
  • When we find what does that mean in the context of the problem? In other words, contextualize this problem, put these symbols back into words.
  • Talk about the findings in part B of the first question. Explain it!
  • In part B of the second problem, explain using the context of the problem what these compositions mean. Which way do most stores apply tax and discounts?


Assign Homework 5 - Basic Functions.  There is a homework quiz over #3-5 planned in two lessons from now (not tomorrow, but the next day). I think this assignment will be challenging for students. There will not be any additional homework tomorrow night so students can take time to figure this assignment out and get help as needed.