ALEX | Alabama Learning Exchange

Building Functions: Composition of Functions

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This lesson provided by:

Author:	Lorie White
System:	Lauderdale County
School:	Rogers High School

General Lesson Information

Lesson Plan ID:	35604
Title:	Building Functions: Composition of Functions
Overview/Annotation:	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.

Associated Standards and Objectives

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability

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

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function	Input	Output
G	Amount of studying: s	Grade in course: G(s)
S	Grade in course: g	Amount of screen time: S(g)
T	Amount of screen time: t	Number of follers: T(t)

Unpacked Content

Evidence Of Student Attainment:

Students:
Given different types of standard functions

Use arithmetic operations to combine functions in context.
Use function composition to combine functions in context.
Write, evaluate, and interpret combined functions in context.

Teacher 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.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
ALGI.17.1: Define functions, relations (ordered pairs), input, output.
ALGI.17.2: Recall how to complete input/output tables.
ALGI.17.3: Recall how to read/interpret information from a table.
ALGI.17.4: Identify algebraic expressions.
ALGI.17.5: Recall how to name points from a graph (ordered pairs).
ALGI.17.6: Recall how to name points on a Cartesian plane using ordered pairs.

a.
ALGI.17.7: Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
ALGI.17.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.

b.
ALGI.17.9: Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtration, and multiplication.

Prior Knowledge Skills:

Explain the distributive property.
Give examples of the properties of operations including distributive.
Combine like terms of a given expression.
Recognize the correct order to solve expressions with more than one operation.
Calculate a numerical expression (Ex. V=(4x4x4).
Choose the correct value to replace each variable in the algebraic expression (Substitution).
Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).

Local/National Standards:

Primary Learning Objective(s):

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.

Additional Learning Objective(s):

Preparation Information

Total Duration:	31 to 60 Minutes
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
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.

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 functions 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 given an opportunity 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.

Assessment

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 around 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:

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 provided extra instruction for students needing help.

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

View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.

ALEX Lesson Plan

Building Functions: Composition of Functions