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Unit 8, Lesson 10

Domain and Range (Part 1)

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Integrated Math 1

10.1

Warm-up

5 mins

Number of Barks

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

This Warm-up prompts students to consider possible input and output values for a familiar function in a familiar context. The work here prepares students to do the same in other mathematical contexts and to think about domain and range in the rest of the lesson.

Launch

None

Activity

None

Student Task Statement

The total number of times a dog has barked is a function of the time, in seconds, after its owner tied its leash to a post and left. Less than 3 minutes after he left, the owner returned, untied the leash, and walked away with the dog.

Could each value be an input of the function? Be prepared to explain your reasoning.

15

300
Could each value be an output of the function? Be prepared to explain your reasoning.

15

300

Student Response

to access Student Response.

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Invite students to share their responses and reasoning. Highlight explanations that make a convincing case for why values beyond 180 could not be inputs for this function and why fractional values could not be outputs.

Some students may argue that 300 could be an input because "300 seconds after the dog's owner walked away" is an identifiable moment, even though the dog and its owner have walked away and may no longer be near the post. Acknowledge that this is a valid point, and that it highlights the need for a function to be more specifically defined in terms of when it "begins" and "ends." If time permits, solicit some ideas on how this could be done.

Tell students that in this lesson, they will think more about values that make sense as inputs and outputs of functions.

10.2

Activity

20 mins

Card Sort: Possible or Impossible?

Instructional Routines

Card Sort MLR8: Discussion Supports

Materials

To Copy (from Blackline Masters)

Possible or Impossible Cards

Activity Narrative

The goal for this activity is for students to define the domain of a function by examining what inputs are possible for various functions. The activity also gives students different types of numbers that they may consider when examining domains of functions in the future.

Students sort different values as inputs during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the values, but especially for categories that distinguish between different number types such as rationals, positives, negatives, and integers.

Each blackline master contains two sets of cards. Here are the numbers on the cards for your reference and planning:

-3
9
15
0.8
4
0
0.001
-18
6.8
72

As students sort the cards and discuss their thinking in groups, listen for their reasons for classifying a number one way or another. Identify students who can correctly and clearly articulate why certain numbers are or are not possible inputs.

10.3

Activity

10 mins

What about the Outputs?

Instructional Routines

None

Materials

None

Activity Narrative

Earlier, students learned that the domain of a function refers to the set of all possible inputs. In this activity, students are introduced to the range of a function and examine it in terms of a situation. They begin to consider how the domain and range of a function are related to the features of its graph.

Launch

Keep students in groups of 2–4. Give students a few minutes of quiet work time and then a moment to share their responses with their group. Leave a few minutes for whole-class discussion.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select either the area function or the revenue function to analyze.
Supports accessibility for: Organization, Social-Emotional Functioning

Activity

None

10.4

Activity

Optional

10 mins

What Could Be the Trouble?

Instructional Routines

Graph It

Materials

None

Activity Narrative

This optional activity is an opportunity to reason about domain more abstractly. Students evaluate an expression that defines a function at some values of input and notice a value that produces an undefined output. They graph the function to examine its behavior and then think about how to describe the domain of the function. Rational functions are explored in more depth in a later course.

Launch

A device with access to graphing technology is needed for every student.

Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use graphing technology, some students may benefit from a checklist or access to step-by-step instructions.
Supports accessibility for: Organization, Memory, Attention

Lesson Synthesis

Tell students that function gives the number of minutes a person sleeps as a function of the number of hours they sleep in a 24-hour period. Display a graphic organizer such as shown.

	in the domain?	in the range?
negative values
0
between 0 and 1
24
25
60
nonnegative fractions
values greater than 480
1,500

Ask students to decide whether each value or set of values described in the first column could be in the domain and in the range of the function. They should be prepared to explain their decisions (some of which may depend on the assumptions they made about the situation).

Once the class completes the organizer (an example is shown here), give students a moment to come up with a holistic description of the domain and range of this function.

	in the domain?	in the range?
negative values	no	no
0	yes	yes
between 0 and 1	yes	yes
24	yes	yes
25	no	yes
60	no	yes
nonnegative fractions	yes	yes
values greater than 480	no	yes
1,500	no	no

to access Cool-down.

Student Lesson Summary

The domain of a function is the set of all possible input values. Depending on the situation represented, a function may take all numbers as its input or only a limited set of numbers.

Function gives the area of a square, in square centimeters, as a function of its side length, , in centimeters.
- The input of can be 0 or any positive number, such as 4, 7.5, or . It cannot include negative numbers because lengths cannot be negative.
- The domain of includes 0 and all positive numbers (or ).
Function gives the number of buses needed for a school field trip as a function of the number of people, , going on the trip.
- The input of can be 0 or positive whole numbers because a negative or fractional number of people doesn’t make sense.
- The domain of includes 0 and all positive whole numbers. If the number of people at a school is 120, then the domain is limited to all non-negative whole numbers up to 120 (or ).
Function gives the total number of visitors to a theme park as a function of days, , since a new attraction opened to the public.
- The input of can be positive or negative. A positive input means days since the attraction opened, and a negative input means days before the attraction opened.
- The input can also be whole numbers or fractional. The statement refers to 17.5 days after the attraction opened.
- The domain of includes all numbers. If the theme park had opened exactly one year before the new attraction opened, then the domain would be all numbers greater than or equal to -365 (or ).

The range of a function is the set of all possible output values. Once we know the domain of a function, we can determine the range that makes sense in the situation.

The output of function is the area of a square in square centimeters, which cannot be negative but can be 0 or greater, not limited to whole numbers. The range of is 0 and all positive numbers.
The output of is the number of buses, which can only be 0 or positive whole numbers. If there are 120 people at the school, however, and if each bus could seat 30 people, then only up to 4 buses are needed. The range that makes sense in this situation would be any whole number that is at least 0 and at most 4.
The output of function is the number of visitors, which cannot be fractional or negative. The range of , therefore, includes 0 and all positive whole numbers.

Launch

Arrange students in groups of 2, and distribute the pre-cut cards. Allow students to familiarize themselves with the values on the cards:

Give students 1 minute to sort the cards into categories of their choosing.
Pause the class after students have sorted the cards.
Select groups to share their categories and how they sorted their cards.
Discuss as many different types of categories as time allows.

Attend to the language that students use to describe their categories and values, giving them opportunities to describe their values more precisely. Highlight the use of terms like “positive,” “negative,” “whole numbers,” “integers,” and “rational numbers.” After a brief discussion, invite students to complete the remaining questions.

Clarify that the cards will get sorted three times (once for each function), so students should record their sorting results for one function before moving on to the next function.

Some students may be unfamiliar with camps and may not know that units other than degrees Fahrenheit and Celsius are used to measure temperature. Provide a brief orientation, if needed.

MLR8 Discussion Supports. Display sentence frames to help groups respond to each other: “_____ is a possible/impossible input because . . . .” and “I agree/disagree because . . . .” When monitoring discussions, revoice student ideas to demonstrate mathematical language. This will help students listen and respond to each other as they explain how they sorted the cards.
Advances: Speaking, Representing

Activity

None

Student Task Statement

Your teacher will give you a set of cards that each contain a number. Decide whether each number is a possible input for the functions described here. Sort the cards into two groups—possible inputs and impossible inputs. Record your sorting decisions.

The area of a square, in square centimeters, is a function of its side length, , in centimeters. The equation defines this function.
1. Possible inputs:
2. Impossible inputs:
A tennis camp charges $40 per student for a full-day camp. The camp runs only if at least 5 students sign up, and it limits the enrollment to 16 campers a day. The amount of revenue, in dollars, that the tennis camp collects is a function of the number of students that enroll.

The equation defines this function.
1. Possible inputs:
2. Impossible inputs:
The relationship between temperature in degrees Celsius and the temperature in kelvin can be represented by a function . The equation defines this function, where is the temperature in degrees Celsius and is the temperature in kelvin.
1. Possible inputs:
2. Impossible inputs:

Student Response

to access Student Response.

Building on Student Thinking

Are You Ready for More?

Student Task Statement

Consider the function .

To find out the sets of possible input and output values of the function, Clare created a table and evaluated at some values of . Along the way, she ran into some trouble.

Find for each -value Clare listed. Describe what Clare’s trouble might be.

-10 0 2 8
Use graphing technology to graph function . What do you notice about the graph?
Use a calculator to compute the value you and Clare had trouble computing. What do you notice about the computation?
How would you describe the domain of function ?

	-10	0		2	8

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Lesson 10

Domain and Range (Part 1)

Warm-up

Number of Barks

Standards Alignment

Building On

Addressing

HSF-IF.B.5

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Card Sort: Possible or Impossible?

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Activity

What about the Outputs?

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity

What Could Be the Trouble?

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.B.5

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.B

HSF-IF.B.5

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.B.5

Building Toward