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Unit 2, Lesson 5

Equations and Their Graphs (Optional)

Algebra 1

5.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Which Three Go Together?

Materials

None

Activity Narrative

This Warm-up prompts students to carefully analyze and compare features of graphs of linear equations. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and to learn how they talk about characteristics of graphs.

The work here prepares students to reason about solutions to equations by graphing, which is the focus of this lesson.

Launch

Arrange students in groups of 2–4. Display the graphs for all to see.

Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask students to share their reasoning as to why a particular graph does not belong, and together to find at least one reason that each item doesn't belong.

Activity

None

Which three go together? Why do they go together?

<p>A line graphed on a set of axes, origin O. Hours versus dollars. The line starts at O and extends up and to the right at a somewhat steep angle.</p> — A

<p>Graph of a line. Vertical axis, hours. Horizontal axis, dollars.</p> — B

<p>Graph of 12 plotted points. Vertical axis, dollars. Horizontal axis, hours.</p> — C

<p>Graph of a line. Vertical axis, dollars. Horizontal axis, hours.</p> — D

Student Response

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Building on Student Thinking

Activity Synthesis

Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology that they use, such as "y-intercept" or "negative slope," and to clarify their reasoning as needed. Consider asking:

“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”

5.2

Activity

Instructional Routines

MLR6: Three Reads

Materials

None

Activity Narrative

Previously, students saw that an equation in two variables can have many solutions because there are many pairs of values that satisfy the equation. This activity illustrates that idea graphically. Students see that the coordinates of all points on the graphs are pairs of values that make the equation true, which means that they are all solutions to the equation.

They also see that, because the given equation models the quantities and constraints in a situation, not all points on the graph are meaningful. For example, only positive or values on the graph (that is, only points in the first quadrant of the coordinate plane) have meanings in this context, because almonds and figs cannot have negative values for their weight.

During the activity, look for students who perform numerical computations straightaway and those who first write a variable equation and then use it to answer the first two questions.

This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

5.3

Activity

Instructional Routines

Graph It MLR7: Compare and Connect

Materials

To Gather

Graphing technology

Activity Narrative

In the previous activity, students analyzed and interpreted points on a graph relative to an equation and a situation. In this activity, they write a linear equation to model a situation, use graphing technology to graph the equation, and then use the graph to solve problems. Each given situation involves an initial value and a constant rate of change.

Before students begin the activity, introduce them to the graphing technology available in the classroom. Offer a quick tutorial on how to graph equations, adjust the graphing window, and plot points. This tutorial could happen independently of the activity as long as it precedes the activity.

Lesson Synthesis

To help students sum up the key ideas of the lesson, display dynamic graphs of the two equations from the "Graph It!" activity. Also, display these questions for all to see:

In the first situation, will the student have \$5,000 after saving for 20 weeks?
In the second situation, after how many minutes should we stop the draining if we want to leave 150 gallons of water in the tank?

Discuss with students:

"How can we use the graph of to answer the first question?" (See if the point is on or below the graph. It is not, so the answer is no.)
"How can we use the graph of to answer the second question?" (Find the point on the graph where the -value is 150, and see what the -value is.)
"On the graph for the first situation, what does the point mean?" (After 15 weeks, there is \$3,000 in the bank account.)
"Is that ordered pair a solution to the equation ? How can we tell?" (No. The point is above the graph, not on the graph. After 15 weeks, there is less than \$3,000 in the account.)
"Is a solution to the equation ? Why or why not?" (No. The point is not on the graph. A negative -value also has no meaning in this situation. The point means that after 25 minutes, there are -50 gallons of water in the tank, which doesn't make sense.)
In general, how can a graph help us find solutions to two-variable equations?" (Any point on the graph of the equation is a solution to that equation.)

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Student Lesson Summary

Like an equation, a graph can give us information about the relationship between quantities and the constraints on them.

Suppose we are buying beans and rice to feed a large gathering of people, and we plan to spend \$120 on the two ingredients. Beans cost \$2 a pound and rice costs \$0.50 a pound. If represents pounds of beans and pounds of rice, the equation can represent the constraints in this situation.

The graph of shows a straight line.

<p>Graph of a line. Vertical axis, pounds of rice. Horizontal axis, pounds of beans.</p>

Each point on the line is a pair of - and -values that makes the equation true and is, thus, a solution. It is also a pair of values that satisfy the constraints in the situation.

The point is on the line. If we buy 10 pounds of beans and 200 pounds of rice, the cost will be , which equals 120.
The points and are also on the line. If we buy only beans—60 pounds of them—and no rice, we will spend \$120. If we buy 45 pounds of beans and 60 pounds of rice, we will also spend \$120.

What about points that are not on the line? They are not solutions because they don't satisfy the constraints, but they still have meaning in the situation.

The point is not on the line. Buying 20 pounds of beans and 80 pounds of rice costs , or 80, which does not equal 120. This combination costs less than what we intend to spend.
The point means that we buy 70 pounds of beans and 180 pounds of rice. It will cost , or 230, which is over our budget of 120.

Launch

Arrange students in groups of 2 and provide access to calculators. Invite students to share what snack foods they take on trips. If students are unfamiliar with figs and almonds, share that these foods come from trees that are native to Iran and surrounding countries.

Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and image, without revealing the questions.

In the first read, students read the problem with the goal of comprehending the situation.
For the first read, ask a student to read the problem aloud while everyone else reads with them, and then ask, “What is this situation about? What is going on here?” Allow 1 minute to discuss with a partner and then share with the whole class. A typical response may be “Clare is buying two kinds of food at the store.” Listen for and clarify any questions about the context.
In the second read, students analyze the mathematical structure of the story by naming quantities.
Invite students to read the problem aloud with their partner, or select a different student to read to the class, then prompt students by asking, “What can be counted or measured in this situation?” Give students 30 seconds of quiet think time, followed by another 30 seconds to share with their partner. A typical response may be: “the cost of almonds and figs, the number of pounds purchased of each, or the total before tax”.
In the third read, students brainstorm possible starting points for answering the questions.
Invite students to read the problem aloud with their partner, or select a different student to read to the class. After the third read, reveal the first question, which asks how many pounds of figs she bought, and ask, “What are some ways we might get started on this?” Instruct students to think of ways to approach the question without actually solving it. Give students 1 minute of quiet think time followed by another minute to discuss with their partner. Invite students to name some possible strategies referring to quantities from the second read. Provide these sentence frames as partners discuss: “To relate the cost and amounts of each food, I would . . . .” “Because I know the amount of almonds purchased, I know . . . .” “The amount she has left to spend on figs is . . . because . . . .”
As partners are discussing their solution strategies, select 1–2 students to share their ideas with the whole class. As students are presenting their strategies to the whole class, create a display that summarizes starting points for each question. (Stop students as needed before they share complete solutions or answers.)

Give students time to complete the rest of the activity, and follow that with a whole-class discussion.

Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language

Activity

None

To get snacks for a class trip, Clare went to the “bulk” section of the grocery store, where she could buy any quantity of a product to get the exact amount she needed.

Clare purchased some salted almonds at \$6 a pound and some dried figs at \$9 per pound. She spent \$75 before tax.

If she bought 2 pounds of almonds, how many pounds of figs did she buy?
If she bought 1 pound of figs, how many pounds of almonds did she buy?
Write an equation that describes the relationship between pounds of figs and pounds of almonds that Clare bought, and the dollar amount that she paid. Be sure to specify what the variables represent.
Here is a graph that represents the quantities in this situation.

Graph of 2 intersecting lines and 5 points, origin O, with grid. Horizontal axis, almonds (pounds), scale is 0 to 15 by 1’s. Vertical axis, dried figs (pounds), scale is 0 to 10 by 1’s. Line AB passes through point A 2 comma 7, point C 6 point 5 comma 4 and point B 11 comma 1. Horizontal line E passes through the origin and point E 14 comma 0. Point D 1 comma 1.
1. Choose any point on the line, state its coordinates, and explain what it tells us.
2. Choose any point that is not on the line, state its coordinates, and explain what it tells us.

Student Response

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Building on Student Thinking

Some students may say that the points not on the line are impossible given that Clare spent $75. Encourage these students to think about what those points would mean if we didn’t know how much money Clare spent.

Launch

Give all students access to graphing technology. Tell students that in this course they will frequently use technology to create a graph that represents an equation and use the graph to solve problems.

Demonstrate how to use the technology available in your classroom to create and view graphs of equations. Explain how to enter equations, adjust the graphing window, and plot a point. If using Desmos, please see the digital version of this activity for suggested instructions.

Arrange students in groups of 2–4. Assign one situation to each group. Ask students to answer the first few questions, including writing an equation, and then graph the equation and answer the last question.

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 demonstration or access to step-by-step directions for how to enter equations, adjust the graphing window, and plot a point.
Supports accessibility for: Organization, Memory, Attention

Activity

None

A student has a savings account with \$475 in it. She deposits \$125 of her paycheck into the account every week. Her goal is to save \$7,000 for college.
1. How much will be in the account after 3 weeks?
2. How long will it take before she has \$1,350?
3. Write an equation that represents the relationship between the dollar amount in her account and the number of weeks of saving.
4. Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3 weeks and the week she has \$4,000. Write down the coordinates.
5. How long will it take her to reach her goal?
A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.
1. How many gallons will be in the tank after 7 minutes?
2. How long will it take for the tank to have 200 gallons?
3. Write an equation that represents the relationship between the gallons of water in the tank and minutes the tank has been draining.
4. Graph your equation using graphing technology. Mark the points on the graph that represent the gallons after 7 minutes and the time when the tank has 200 gallons. Write down the coordinates.
5. How long will it take until the tank is empty?

Student Response

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Audience

Assessment Access

Lesson 5

Equations and Their Graphs (Optional)

Warm-up

Standards Alignment

Building On

8.EE.B

8.F.B.5

Addressing

Building Toward

HSF-IF.C.7.a

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-CED.A.2

HSA-CED.A.3

HSA-REI.D.10

Building Toward

Standards Alignment

Building On

Addressing

HSA-CED.A.2

HSA-CED.A.3

HSA-REI.D.10

Building Toward

HSF-IF.C.7.a