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Unit 7, Lesson 6

Solving Problems with Inequalities in Two Variables

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Integrated Math 1

6.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Graph It

Materials

To Gather

Graphing technology

Activity Narrative

In this Warm-up, students use graphing technology to graph simple linear inequalities in two variables. They practice adjusting the graphing window until the solution regions become visible and give useful information. Later in the lesson, students will write inequalities that represent constraints in different situations and find the solution sets. The exercises here prepare students to do the latter, using graphing technology.

Launch

Give students access to graphing technology. If using Desmos:

Explain to students that typing " " gives the symbol and typing "" gives the symbol.
Remind students that the and buttons can be used to zoom in and out of the graphing window, and that the wrench button in the upper-right corner can be used to set the graphing window precisely.

If using other graphing technology available in your classroom:

Demonstrate how to enter the and symbols.
Remind students how to set a useful graphing window by zooming in or out, and how to set a precise graphing window by specifying the horizontal and vertical boundaries.
(For technology that takes equations or inequalities in slope-intercept form only.) Remind students that some inequalities might need to be rewritten such that is isolated before the inequality can be entered into the graphing tool.

Activity Synthesis

Display the correct solution regions for all to see, and ask students to check their graphs. Discuss any challenges that students may have come across when trying to graph using technology.

Explain to students that they will now use graphing technology to find solutions to some inequalities that represent constraints in situations.

6.2

Activity

Instructional Routines

Aspects of Mathematical Modeling Graph It

Materials

To Gather

Graphing technology

Activity Narrative

This activity enables students to integrate several ideas and skills from the past few lessons. Students write inequalities in two variables to represent constraints in situations, use technology to graph the solutions, interpret points in the solution regions, and use the inequalities and the graphs to answer contextual questions. In doing so, they engage in aspects of mathematical modeling (MP4).

The questions in this activity are written in pairs. The same constraints and contexts will be used in an upcoming lesson on systems of linear inequalities in two variables.

Decide on the structure for the activity depending on the time available and the amount of practice each student needs. Here are some possibilities:

Assigning all three pairs of questions to all students.
Assigning each student a pair of questions, arranging students who work on different pairs in groups of 3, and asking them to explain their solutions to one another.
Arranging students in groups of 3, assigning the same pair of questions to each group, and—for each pair of questions—asking one group to present the solutions to the class.

6.3

Activity

Optional

Instructional Routines

Card Sort MLR7: Compare and Connect Take Turns

Materials

To Copy (from Blackline Masters)

Representations of Inequalities Cards

Activity Narrative

This optional activity allows students to practice interpreting inequalities in context and reasoning about their solutions graphically and numerically. A sorting and matching task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

As students explain their thinking to a partner, encourage them to use precise language and mathematical terms to refine their explanations (MP6).

Launch

Lesson Synthesis

Summarize the lesson by discussing students' work for an earlier activity—about the club donations, concert tickets, and advertising packages—and inviting them to reflect on their reasoning process. Discuss questions such as:

"Of the four things you were asked to do in the last activity—writing an inequality, graphing the solutions, identifying and interpreting a particular solution, and answering the question about the situation—which one did you find the most challenging or prone to error?"
"How is graphing linear inequalities using technology similar to graphing them by hand? How is it different?"
"You have previously used technology to graph linear equations in two variables. How is graphing linear inequalities in two variables different from graphing equations?"

to access Cool-down.

Student Lesson Summary

Suppose we want to find the solution to . We can start by graphing the related equation .

When identifying the solution region, it is important not to assume that the solution will be above the line because of a “>” symbol or below the line because of a “<” symbol.

Instead, test the points on the line and on either side of the line, and see if they are solutions.

For , points on the line and above the line are not solutions to the inequality because the pairs make the inequality false. Points that are below the lines are solutions, so we can shade that lower region.

Graphing technology can help us graph the solution to an inequality in two variables.

Many graphing tools allow us to enter inequalities such as and will show the solution region, as shown here.

Some tools, however, may require the inequalities to be in slope-intercept form or another form before displaying the solution region. Be sure to learn how to use the graphing technology available in your classroom.

<p>Inequality graphed on a coordinate plane.</p>

Although graphing using technology is efficient, we still need to analyze the graph with care. Here are some things to consider:

The graphing window. If the graphing window is too small, we may not be able to really see the solution region or the boundary line, as shown here.
The meaning of solution points in the situation. For example, if and represent the lengths of two sides of a rectangle, then only positive values of and (or points in the first quadrant) make sense in the situation.

Launch

Continue to provide access to graphing technology. Explain that students will now write and graph inequalities to solve problems about some situations.

Assign at least one pair of questions about the same context to each student. See the Activity Narrative for some possible ways to structure the activity.

Some students might not be familiar with terms such as "savings," "checking," or "premium." Explain any unfamiliar terms as needed.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select one of the situations to work on. If students are arranged in groups, each member should be prepared to share during the whole-class discussion.
Supports accessibility for: Organization, Social-Emotional Functioning

Activity

None

Here are three situations. There are two questions about each situation. For each question that you work on:

a. Write an inequality to describe the constraints. Specify what each variable represents.

b. Use graphing technology to graph the inequality. Sketch the solution region on the coordinate plane and label the axes.

c. Name one solution to the inequality and explain what it represents in that situation.

d. Answer the question about the situation.

Club Donations

A school environmental club is raising money during the year to donate in support of two causes: a national organization lobbying for clean air and a local non-profit working to restore a stream that runs past the school. They plan to donate a maximum of $600 by the end of the year, some toward clean air and some toward stream restoration.

If the club donates \$200 to stream restoration, what can you say about the amount they donate to clean air?
When the restoration of the stream is complete, the name of any person or group that donates \$500 or more will be added to a commemorative plaque. The national organization lobbying for clean air does not offer any donation perks.

If the club donates enough to get its name on the plaque, what can you say about the amount it donated to each group?

Concert Tickets

Two kinds of tickets to an outdoor concert were sold: lawn tickets and seat tickets. Fewer than 400 tickets in total were sold.

If you know that exactly 100 lawn tickets were sold, what can you say about the number of seat tickets?
Lawn tickets cost \$30 each, and seat tickets cost \$50 each. The organizers want to make at least $14,000 from ticket sales.

If you know that exactly 200 seat tickets were sold, what can you say about the number of lawn tickets?

Advertising Packages

An advertising agency offers two packages for small businesses that need advertising services. A basic package includes only design services. A premium package includes design and promotion. The agency's goal is to sell at least 60 packages in total.

If the agency sells exactly 45 basic packages, what can you say about the number of premium packages it needs to sell to meet its goal?
The basic advertising package has a value of \$1,000 and the premium package has a value of \$2,500. The goal of the agency is to sell more than $60,000 worth of small-business advertising packages.

If you know that exactly 10 premium packages were sold, what can you say about the number of basic packages that the agency needs to sell to meet its goal?

Student Response

to access Student Response.

Building on Student Thinking

One inequality in the bank account context involves only one variable. Some students might think that all inequalities they write must include two variables. Reassure them that this might not always be the case. Consider pointing to examples from earlier activities in which they wrote or graphed inequalities such as or .

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Lesson 6

Solving Problems with Inequalities in Two Variables

Warm-up

Standards Alignment

Building On

Addressing

HSA-REI.D.12

Building Toward

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-CED.A.3

HSA-REI.D.12

Building Toward

Standards Alignment

Building On

Addressing

HSA-CED.A.3

HSA-REI.D.12

Building Toward