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Unit 6, Lesson 9

Interpreting Functions

Extra Support Materials for Algebra 1

9.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

The purpose of this Warm-up is to elicit the idea that graphs can be discrete or continuous, which will be useful when students interpret exponential functions and make sense of whether a graph should be discrete or continuous in the associated Algebra 1 lesson. While students may notice and wonder many things about these graphs, the fact that one is a line and one is a discrete set of points are the important discussion points.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Launch

Arrange students in groups of 2. Display the table and the 2 graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

Here is a table of values of data that was collected and two graphs of the data. What do you notice? What do you wonder?

	0	1	2	3	4	5	6
	6	5	4	3	2	1	0

<p>Scatter plot from 0 comma 6 to 6 comma 0. As x increases by 1, y decreases by 1 for all points in between </p>

<p>Line with x and y intercepts of 6</p>

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the graphs. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the situation being represented does not come up during the conversation, ask students to discuss this idea. What are some situations that might be appropriate to represent with just points, versus a connected line? (Students will see many examples in this lesson, so it’s not necessary to dwell on this question for long.)

9.2

Activity

Instructional Routines

Aspects of Mathematical Modeling

Materials

None

Activity Narrative

The purpose of this activity is to elicit the idea that graphs can be discrete or continuous, depending on the context, which will be useful when students interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson. When students pay close attention to the appropriate domain and range, both restricted by the context, they are engaging in work that is important in modeling with mathematics (MP4). Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange the students in groups of 4. Each member of the group should choose one of the questions and work individually. After quiet work time, ask students to share their responses within their group and decide if they are correct. Follow with a whole-class discussion.

In order to find the table of values, students may choose to use graphing technology, if available.

Representation: Develop Language and Symbols. Make connections between representations visible. Annotate connections between the verbal description, table, and graph.
Supports accessibility for: Language, Conceptual Processing

9.3

Activity

Instructional Routines

Aspects of Mathematical Modeling MLR2: Collect and Display

Materials

None

Activity Narrative

Given a description of a context, students determine the restrictions on the domain and range imposed by the context. Considering how a context restricts the domain or range of a function modeling it is an important part of mathematical modeling (MP4). This prepares students to interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson.

This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.

Launch

Remind students that domain refers to possible values of the independent variable, and range refers to possible values of the dependent variable. Use a few examples from the previous activity to illustrate. For example, for the cab ride, the domain is all numbers that are greater than 0, and the range is all numbers that are greater than 3.5. This presumes that a ride must have some distance (it can’t be 0 miles long), and can be any number of miles long. A modeler might decide not to consider any cab rides that are longer than, say, 100 miles, in which case the domain would be all numbers between 0 and 100. Sometimes restrictions on the domain are a decision made by the modeler.

In contrast, for the function modeling the admission price to the park based on number of people, the domain contains only positive whole numbers, since the number of people must be a positive whole number.

Arrange students in groups of 2. After a few minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they look different. Follow with a whole-class discussion.

Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe domain and range. Display words and phrases, such as “domain,” “range,” “independent,” “dependent,” “positive,” “negative,” “whole numbers,” “fractions,” and “decimals.”

Here are descriptions of relationships between quantities. For each description:

Make a table of at least 5 pairs of values that represent the relationship.
Plot the points. Label the axes of the graph.
Answer these questions and explain your reasoning: Should the points be connected? Are there any input or output values that don’t make sense?

A cab charges \$1.50 per mile plus \$3.50 for entering the cab. The cost of the ride is a function of the miles ridden , and is defined by .
The admission to the state park is \$5.00 per vehicle plus \$1.50 per passenger. The total admission for one vehicle is a function of the number of passengers , defined by the equation .
A new species of mice is introduced to an island, and the number of mice is a function of the time in months since they were introduced. The number of mice is represented by the model .
When you fold a piece of paper in half, the visible area of the paper gets halved. The area is a function of the number of folds , and is defined by .

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Audience

Assessment Access

Lesson 9

Interpreting Functions

Warm-up

Standards Alignment

Building On

Addressing

8.F.B.5

Building Toward

HSF-IF.B.5

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Student Response

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

8.F.A.1

8.F.B.5

Building Toward

HSF-IF.B.5

Standards Alignment

Building On

Addressing

8.F.A.1

8.F.B.4

Building Toward

HSF-IF.B.5