Lesson Plan

Building Functions: Composition of Functions

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This lesson provides a review of evaluating functions and finding function rules as well as an introduction to the composition of functions. The review is accomplished through the use of an online exploration using a function machine. The idea of a function machine is also used to explain the composition of functions. Instruction is provided in finding the composition using several different representations of functions (input/output tables, graphs, and function rules).

This lesson results from the ALEX Resource Gap Project.

Content Standards

Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.17
Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

Unpacked Content

Vocabulary

Function composition

Knowledge

Students know:

Techniques to combine functions using arithmetic operations.
Techniques for combining functions using function composition.

Skills

Students are able to:

Accurately develop a model that shows the functional relationship between two quantities.
Accurately create a new function through arithmetic operations of other functions.
Present an argument to show how the function models the relationship between the quantities.

Understanding

Students understand that:

Arithmetic combinations of functions may be used to improve the fit of a model.

Interdisciplinary Connections

Learning Objectives

Primary Learning Objectives

Students will be able to write a function rule given domain and range values.
Students will be able to find the range values of a composition of two functions given the two function rules and a domain value.
Students will be able to write a function rule for the composition of two functions given the function rules for the two functions.
Students will be able to find the value of the composition of two functions given the graphs of the two functions.

Procedures/Activities

Before:

1. Play "What's My Rule": Write a simple equation on a card, for example, (y=x^2-3). The teacher should select 4 students to provide an input value from 1 to 10. The teacher will use mental calculations to provide an output value. Working in pairs, have students attempt to guess the rule. Allow each pair of students to write down their function and prove by substitution that it works.

2. Using computers, direct students to the Function Machine Activity on the Math Playground website. Students will need to select the advanced level. Allow them to practice writing a function rule when given domain and range values. The difficulty of the activity can be adjusted by changing the level (1,2, or 3 found at the bottom of the screen) or by allowing students to select the domain values themselves. This activity will provide an association to a function machine which will be used in the explain portion of the lesson. Allow 5 - 10 minutes for this activity. You may wish to pair students to provide peer tutoring. This could be done as a whole group activity if individual computers are not available.

During: Using the PowerPoint presentation teachers will explain the process of finding a composition of functions.

1. Teachers will display the slide showing a composition of functions using the analogy of a function machine. Show students how an input value is dropped into the machine and an output value falls out into another machine thus producing a composition.

2. Working in pairs students will be prompted to complete the Composition of Functions Notes page for the function machine. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.

3. Teachers will display the slide showing a composition of functions from a table of values. Explain to students how to find the output associated with the given input for the first function and then use that output as the input for the second function to obtain the value for the composition.

4. Working in pairs, students will be prompted to complete the Composition of Functions Notes page for the table of values. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.

5. Teachers will display the slide showing a composition of functions using graphs. Explain to students how to find the output (y-value) associated with the given input (x-value) on the first graph and then allow that output (y-value) to become the input (x-value) for the second function to find the value of the composition.

6. Working in pairs, students will be prompted to complete the Composition of Functions Notes page for graphs. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.

7. Teachers will display the slide showing a composition of functions using function rules. They need to emphasize that this time the function rule itself will be the input for the second function.

8. Working in pairs, students will be prompted to complete the Composition of Functions Notes page for function rules. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.

The notes can be used as an assessment tool to determine if additional examples or instruction is needed. Allow 30 - 40 minutes for instruction.

After: Students will complete a Frayer Model for the composition of functions. Teachers will ask students to complete the form by writing a definition in their own words for the composition of functions. Next, they will provide an example of any kind for the composition of two functions. They will be asked to describe the different representations of functions that were discussed, and finally, they will be allowed to express any problems or concerns that they may have. This will be their ticket out the door. Allow 5 minutes for students to complete.

Learning Activity (Before)

Learning Activity (During)

Learning Activity (After)

Assessment Strategies

The Function Machine Activity can be used to assess student readiness for the lesson. Students can work independently or in pairs to review evaluating functions and finding rules from function tables. The activity has three different levels of difficulty and students can choose the values or the computer will choose values for them. This assessment will be informal as it will provide a self-check for students.

The Composition of Functions Student Notes can be used as an informal assessment tool. Teachers can circulate the room and check student understanding as they work.

The Frayer Model for the Composition of Functions will be used as an exit ticket. This summative assessment will provide teachers with a quick way to check comprehension and will also allow students to let the teacher know what they do not understand.

Acceleration and Intervention

Acceleration

Advanced students can be encouraged to work on higher levels on the Function Machine Activity.

Advanced students can be encouraged to produce a graph of the composition of functions using the input and output from the composition.

Intervention

Students needing extra help can begin with a lower level on Function Machine Activity. Peer tutoring can also be used to help clear up questions or misconceptions.

By monitoring student work on Function of Compositions Student Notes, the teacher will be able to identify misconceptions and provide extra instruction for students needing help.

The Frayer Model for Composition of Functions can be used to identify students who may need extra instruction.

Approximate Duration

Total Duration

31 to 60 Minutes

Background and Preparation

Background/Preparation

Students should be familiar with the terminology associated with functions, such as domain, range, input, and output.

Students should be able to find the range of a function given the domain.

Students should recognize a function from multiple representations, such as tables, graphs, and function rules.

Teachers should make one copy per student of student notes handout, student exit handout, and load the PowerPoint presentation. Teachers should also be familiar with the Function Machine Activity.

Teachers should also preview the PowerPoint presentation to be familiar with the different representations that are presented in the lesson.

Materials and Resources

Teachers:

Computer connected to a projector

Prepare one copy per student in advance of Student Note worksheet and Student Exit worksheet.

Load the PowerPoint presentation on the teacher computer.

Students:

Individual student computers if available.

Student Note Worksheet

Student Exit Worksheet

Technology Resources Needed

Internet connection is required for the Function Machine Activity

Approved Date

2017-06-20

Author

Owner (Author)

mrswhite

Learning Resource Type

Building Functions: Composition of Functions

Subject Area

Grade(s)

Overview

MA19.A1.17 Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

Unpacked Content

Vocabulary

Knowledge

Skills

Understanding

Primary Learning Objectives

Procedures/Activities

Assessment Strategies

Acceleration

Intervention

Approximate Duration

Total Duration

Background and Preparation

Background/Preparation

Materials and Resources

Materials and Resources

Technology Resources Needed

Approved Date

Owner (Author)

MA19.A1.17
Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.