4_SM_EM2_PowerUp_6-9_Grade7Module1Lesson4 by General

LESSON 4

Exploring Graphs of Proportional Relationships Identify proportional relationships represented as graphs. Interpret and make sense of the point (0, 0) in context.

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

EXIT TICKET Name

Date

Pedro earns extra money doing yard work in his neighborhood. He creates a graph showing the number of hours he works and the amount of money he earns. y

Earnings (dollars)

Lesson at a Glance In this lesson, students connect proportional relationships represented in tables to their graphical representations. Students sort tables into two categories, proportional and not proportional, and match each table with its graph. Students generalize the key characteristics of graphical representations of proportional relationships: that the points lie on a line through the origin. This lesson introduces the term constant of proportionality.

Key Question

• How can we determine whether a graph represents a proportional relationship?

Achievement Descriptors

Time (hours)

a. Based on the graph, does the amount of money Pedro earns appear to be proportional to the number of hours he works? Explain how you know. Yes. The amount of money Pedro earns appears to be proportional to the number of hours he works. The graph looks linear, and it appears that a line connecting the points would go through the origin.

7.Mod1.AD5 Interpret the meaning of any point (x,

The point (0, 0) means that Pedro works 0 hours and earns $0.

y) on the graph

of a proportional relationship in terms of the situation, including the points (0, 0) and (1, r), where r is the unit rate. (7.RP.A.2.d)

c. How much money does Pedro earn if he only works 1 hour? Explain how you know. 40 5

7.Mod1.AD3 Identify the constant of proportionality in proportional

relationships. (7.RP.A.2.b)

b. Pedro adds the point (0, 0) to his graph. What does the point (0, 0) mean in this context?

Pedro earns $40 for 5 hours of work.

7.Mod1.AD2 Recognize proportional relationships. (7.RP.A.2.a)

= 8

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EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

Agenda

Materials

Fluency

Teacher

Launch Learn

5 min

30 min

• Graph Match • Analyzing (0, 0)

• Always True sign • Sometimes True sign • Never True sign • Tape

• Revisiting the Water Flow Problem

Students

• Take a Stand

• Table Sort cards (1 set per student group)

Land

10 min

• Graph Match cards (1 set per student group)

Lesson Preparation • Copy and cut out 1 set of Table Sort cards for each student group. • Copy and cut out 1 set of Graph Match cards for each student group. • Create signs labeled Always True, Sometimes True, Never True, and position them in different locations around the room.

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28-Aug-21 8:23:18 AM

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Fluency Graph Points Students graph points to prepare for identifying proportional relationships represented as graphs. Directions: Graph and label each point in the coordinate plane. A(0, 8) y

B(5, 0)

Teacher Note

10 Students may use the Quadrant I removable.

C(4, 7)

A(0, 8)

(

2, 2 1 2

)

F(3, 7.25) C(4, 7)

E(9 2 , 6)

6 5

G(7.75, 4)

(

E 91,6 2

)

D(2, 2 2)

UDL: Action & Expression

F(3, 7.25)

B(5, 0) 1

x 8

9 10

Provide access to digital graphing tools or adapted grid paper to offer alternatives to physical response methods that require precise fine motor skills.

G(7.75, 4)

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EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

Launch

Rice, x (cups) Number of Copies Made, x

Cost, y (dollars)

Green Beans, x (pounds)

250

3 4

1 2

3 3 4

21 2

300

37.50

6.30

Time, x (hours)

Bike Rental Cost, y (dollars)

14.70 10.50

Time, x (hours)

Area of a Square, y

Value of Change Collected, y (dollars)

0.5

1.10

5.15

7.60

1.5

3.50

7 ▸ M1 ▸ TA ▸ Lesson 4 ▸ Table Sort

Side Length of a Square, x

Teacher Note Graphs are used to represent values in context in grade 6. Depending on students’ understanding, a discussion about when graphs are continuous and when individual points are graphed may be helpful to some students before moving on.

• The proportional relationships all have a constant unit rate.

This page may be reproduced for classroom use only.

What I notice:

120

Cost, y (dollars)

1. Sort the tables into two categories: proportional and not proportional.

100

Water, y (cups)

EUREKA MATH2

Divide students into groups of three or four. Distribute one set of Table Sort cards to each group. Have students work together to sort the tables into two categories: proportional and not proportional. As groups finish sorting, encourage them to record anything they notice about the tables in each category.

Students sort tables by examining their characteristics.

• The proportional relationships all have a multiplicative structure.

Consider reminding students that when individual points are graphed, this signifies that only these values have meaning in the context. If the graph is a continuous line, then all points on that line have meaning in the context.

• Relationships that are not proportional may still include the ordered pair (0, 0). When most groups have finished, call the class back together and invite students to share what they noticed about the tables in each category. Facilitate a class discussion by using the following prompts. How did your group determine which tables represented proportional relationships and which did not? When I divide all pairs of values in the table and get the same number, I know the relationship is proportional. When I divide and get different numbers, the relationship is not proportional. In the last two lessons, you identified key features of a proportional relationship represented in a table. What do you think the graphs of proportional relationships look like? Allow time for student responses. Now let’s examine the graphs of proportional relationships. The groupings shown in the table give the solution for the Launch activity as well as the first segment of Learn. Copyright © Great Minds PBC

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Differentiation: Support To succeed in this lesson, students need to know how to find the constant unit rate of a proportional relationship. Use student work on the lessons 2 and 3 Exit Tickets to gauge student understanding. If a student needs additional support, consider analyzing one table. Have the student add a row or column to the table. Calculate the unit rate for each ordered pair in the table and write the unit rate in the added row or column. Compare the values to see whether they are constant.

29-Aug-21 4:30:52 PM

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Proportional

Number of Copies Made, x

100

120

250

300

50 45 40

Cost, y (dollars)

37.50

35 30 25 20 15 10 5 0

100

150

200

250

300

Rice, x (cups)

Water, y (cups)

3 4

1 2

6 5 4 3 2

3 62

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3 4

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EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

Proportional y

Green Beans, x (pounds)

Cost, y (dollars)

10 9

6.30

14.70

10.50

7 6 5 4 3 2 1 0

1.5

2.5

Not Proportional y

Time, x (hours)

Bike Rental Cost, y (dollars)

25 20 15 10 5

24 0

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0.5

3.5

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EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Not Proportional

Side Length of a Square, x Area of a Square, y

9 8

7 6 5 4 3 2 1 0

0.5

1.5

2.5

1.5

3.5

4.5

Time Passed, x (hours)

Value of Change Collected, y (dollars)

7 6 5

0.5

1.10

5.15

7.60

1.5

3.50 0

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0.5

2.5

3.5

21-Aug-21 2:34:25 PM

EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

Learn 7 ▸ M1 ▸ TA ▸ Lesson 4 ▸ Graph Match

This page may be reproduced for classroom use only.

Students examine graphs of proportional relationships and generalize about their characteristics.

Graph Match

Promoting the Standards for Mathematical Practice y

y 50

45 40

30 25

15 10

5 0

100

150

200

250

300

y 10

Distribute a set of Graph Match cards to each group. Tell students to match each graph card to one of the tables they sorted into proportional and not proportional categories. Each graph has exactly one matching table.

7 6

Ask the following questions to promote MP8:

15 5 4

3 2

1 0

0.5

1.5

2.5

3.5

• What patterns did you notice when you looked at the graphs of proportional relationships?

8 6

7 5

6 5

3 2

2 1

1 0

0.5

1.5

2.5

3.5

4.5

0.5

1.5

2.5

3.5

EUREKA MATH2

Circulate as groups work and encourage students to record anything they notice about the graphs of proportional relationships. Identify students who recognize that the graphs of proportional relationships begin at the origin, (0, 0), and lie on a line.

Students look for and express regularity in repeated reasoning (MP8) when they look at examples of graphs and observe key features of graphs representing proportional relationships.

• What is the same about the graphs of the proportional relationships?

2. Match each graph to its table. Examine the graphs of the proportional relationships. What characteristics do they have? What I notice: • T he graphs of the proportional relationships are lines or are points that lie on the same line. • The graphs of the proportional relationships go through the origin. When most groups have finished, call the class back together and facilitate a class discussion by asking the following questions. What strategies did your group use to match the graphs to the tables? We graphed the values from the table and then found the graph card that matched. We identified a few points on each graph and then looked to see which table had the same pairs of numbers.

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EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

What characteristics do graphs of proportional relationships have in common? Have several groups share the characteristics they identified. Encourage the students who noticed that the graphs of proportional relationships begin at the origin, (0, 0), and lie on a line to share with the whole class. Allow time for students to record what classmates noticed to problem 2 as needed. If no students identified the properties of the graphs of proportional relationships, ask questions like the following to guide students to this understanding. How would you describe the graphs that represent proportional relationships? The graph is either a line or points that appear to lie on a line. Do the graphs of proportional relationships always have to appear as lines or can they be points that appear to lie on a line too? Can both types of graphs represent proportional relationships? Both types of graphs can represent proportional relationships. To represent a proportional relationship, the x- and y-values plotted on the graph must reflect a constant unit rate. Do all graphs that are lines represent proportional relationships? No. The graph of the relationship between the number of hours and the bike rental cost formed a line. However, the table showed that it was not proportional because the pairs of values did not reflect a constant unit rate.

Teacher Note If students do not notice the similarities in the graphs, encourage them to note the specific x- and y-values of each point. Have them consider the following questions: • Are there any points that are common among all the graphs of proportional relationships? • How do the x- and y-values in each graph change in relation to each other? If students solidify an understanding that the graphs of proportional relationships are lines that include the origin, extend that thinking by asking them to consider unknown values in the situation. Use the following questions to guide their thinking: • Is there an ordered pair that you think might fit into this relationship? • Is there an ordered pair that you know would not fit into this relationship?

What makes the bike rental graph different from the ones that we identified as proportional? The graphs of the proportional relationships all include the point (0, 0), while this graph does not. Do you think a graph could contain the point (0, 0) but not represent a proportional relationship? Yes. The graph that represents the relationship between the side length of a square and its area includes the point (0, 0), but it is not proportional because the graph is not a line.

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UDL: Representation Consider pausing after identifying the characteristics of graphs of proportional relationships. Ask students to think about and visualize lines that do and do not contain the point (0, 0). This supports comprehension about the key characteristics of graphs of proportional relationships.

29-Aug-21 4:34:13 PM

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

What are the characteristics of the graph of a proportional relationship? The graph of a proportional relationship is a line, or points that lie on the same line, that includes the point (0, 0).

Analyzing (0, 0)

Students explain the meaning of the point (0, 0) in context. Allow time for groups to complete the Analyzing (0, 0) problem. 3. Review each relationship that your group identified as proportional. What does the point (0, 0) mean in each context? The cost to make 0 copies is 0 dollars. Making 0 cups of rice requires 0 cups of water. Buying 0 pounds of green beans costs 0 dollars. When students are finished, invite them to share their responses with the class. Now that we have matched tables to graphs that represent proportional relationships, we see that the line containing the graph of a proportional relationship must always include the point (0, 0) because without it the relationship wouldn't have a constant unit rate.

Revisiting the Water Flow Problem Students verify that the graph of a context they know to be proportional fits their description of the graph of a proportional relationship.

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EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Have students work together in their groups for a few minutes to solve problems 4 and 5. Circulate to identify a student who correctly graphs the data from the table so the graph can be displayed during the discussion that follows. In lesson 2, you identified a proportional relationship between the number of minutes a faucet is turned on and the number of gallons of water that flow from it. Time, t (minutes)

Water, g (gallons)

1.5

7.5

Differentiation: Challenge If groups finish early, have them return to other examples of relationships that are proportional or not proportional from lessons 2 and 3. Have students graph the relationships represented in the tables and reflect on the completeness of their description of a proportional graph.

4. Graph the proportional relationship by using the data in the table. g

8 7

Water (gallons)

6 5 4 3 2 1

Time (minutes)

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EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

5. Does the graph fit our description of a proportional relationship? Explain why or why not. Yes, this graph fits our description of a proportional relationship because the points lie on a line and the line passes through the origin (0, 0). Call the class back together, and ask a student to display a graph for the class. Then ask the following questions. Does this graph fit our description of the graph of a proportional relationship? Yes. The points lie on a line, and the line passes through the origin (0, 0). What does the point (0, 0) mean in this context?

The point (0, 0) means that when no time has passed, no water has flowed out of the faucet. In lessons 2 and 3, we verified that the relationship was proportional by looking for a constant unit rate in the table. What is the constant unit rate in this situation? Where can we see this constant unit rate in the graphical representation? Since the constant rate is 1.5 gallons per minute, the constant unit rate is 1.5. In the graph, you can see this as the point (1, 1.5).

The constant unit rate, which we saw in the table and the graph, is called the constant of proportionality. In this situation, the constant of proportionality is 1.5. Considering the context of flowing water, can we draw a line through these points? Yes. It makes sense to connect the points because the water keeps flowing at this rate over time.

Teacher Note Students build an understanding of the term constant of proportionality over the next few lessons. This is an informal description that can guide their learning.

Draw a line through the points on the graph that is displayed, and ask students to do the same on their graph. Will this line extend beyond the first quadrant? No. Negative values for time or volume don’t make sense in this situation.

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EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Take a Stand Students summarize the learning of the day by considering what they know to be true about the graphs of proportional relationships. Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom: Always True, Sometimes True, and Never True. Present the statement “Graphed lines represent proportional relationships.” Then invite students to stand beside the sign that best describes their thinking. When all students are standing near a sign, allow 1 minute for groups to discuss the reasons why they chose that sign. After a minute, ask each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group. Ensure that all students understand that the statement is sometimes true. If the line passes through the origin, it represents a proportional relationship; otherwise, the line does not represent a proportional relationship. Have students return to their seats and respond to problem 6. 6. Is the statement “Graphed lines represent proportional relationships” always, sometimes, or never true?

Teacher Note Although the discussion should land on recognizing that graphed lines sometimes represent proportional relationships, make sure to acknowledge the correct thinking of other rationales. For example, some students may justify standing by Always True because proportional relationships are always represented by lines. This rationale signifies that the student is misinterpreting the statement but is demonstrating a correct understanding of the characteristics of the graph of a proportional relationship. Similarly, some students may justify standing by Never True because proportional relationships include only nonnegative values. This means that the graphed representation is a line, beginning at the origin. This rationale also signifies a complete understanding of the content.

It is sometimes true because lines that do not go through the origin do not represent proportional relationships.

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29-Aug-21 4:36:14 PM

EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

Land Debrief

Language Support

5 min

Objectives: Identify proportional relationships represented as graphs. Interpret and make sense of the point (0, 0) in context. Guide a discussion about the graphs of proportional relationships. How do we know when a graph represents a proportional relationship? A graph that represents a proportional relationship is a line or points that lie on a line.

As the module progresses, students continue to build on their understanding of the term constant of proportionality. As students add to their Proportional Relationships Graphic Organizers, direct them to label the constant unit rate in a proportional relationship as the constant of proportionality. Tell students that they will add to this understanding throughout the module.

A graph that represents a proportional relationship begins at the origin. Have students make additions or revisions to their Proportional Relationships Graphic Organizers. What new information did you add to your graphic organizer about the term proportional relationship? What edits did you make to what was already there? Expect students to add information about the term constant of proportionality and the characteristics of graphs that represent proportional relationships.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Recap

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

RECAP Name

Date

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

2. Consider the following graph for a lemonade recipe. y

This point is (10, 5), and it represents that for every 10 cups of water, there are 5 cups of lemon juice.

Exploring Graphs of Proportional Relationships 8

In this lesson, we represented proportional relationships graphically. identified graphs of proportional relationships as lines that pass through the origin, the point (0, 0).

Lemon Juice (cups)

• •

Terminology The constant of proportionality is the constant unit rate in a proportional relationship between two quantities.

Examples

x 1 2 3 4 5 6 7 8 9

The graph represents a proportional relationship. The graph forms a line that passes through the origin, (0, 0).

9 8 7 6 5 4 3 2 1 0

x 1 2 3 4 5 6 7 8 9

The graph does not represent a proportional relationship. The graph forms a line, but the line does not pass through the origin, (0, 0).

9 8 7 6 5 4 3 2 1 0

Water (cups)

a. Does the graph appear to represent a proportional relationship between the number of cups of water and the number of cups of lemon juice used in a lemonade recipe? Explain how you know. The graph appears to represent a proportional relationship. The points appear to lie on a line that passes through the origin, (0, 0).

x 1 2 3 4 5 6 7 8 9

The graph does not represent a proportional relationship. The graph passes through the origin, (0, 0), but it does not form a line.

8 Lemon Juice (cups)

1. Which graphs represent a proportional relationship? Explain how you know.

9 8 7 6 5 4 3 2 1

Confirm that the points appear to lie on a line that passes through the origin

by sketching a line.

x 0

Water (cups)

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R E CA P

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EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

b. Create a table of values based on the graph. Water (cups)

Lemon Juice (cups)

c. Use the values from the table to justify that the relationship between the number of cups of water and the number of cups of lemon juice is proportional.

The point (10, 5) is represented in the table because the graph shows that for every 10 cups of water, there are 5 cups of lemon juice.

The table shows that there is a constant unit rate, or a constant of proportionality, associated with the relationship between the number of cups of water and the number of cups of lemon juice. The constant of proportionality is 0.5.

To confirm that the relationship is proportional, compare the unit rates for each ordered pair to see whether they are the same number. 1 2

3 5 = 2 = = = 0.5 4

Since the unit rate is 0.5, the constant of proportionality is also 0.5.

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R E CA P

6/7/2020 10:44:38 AM

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

PRACTICE Name

Date

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

4. Consider the given table.

For problems 1–3, determine whether each graph represents a proportional relationship. Explain how you know. y

1 0

The graph does not represent a proportional relationship. The line does not pass through the origin.

a. Graph the relationship. y

1 x

6 5

The graph does not represent a proportional relationship. The graph is a curve, not a line.

4 3 2 1

x 0

10 11 12 13

b. Is the relationship between y and x proportional? Justify your thinking by using the table and graph.

6 5

The relationship between y and x is proportional. The pairs of values from the table reflect a constant unit rate of 13 . The points on the graph appear to lie on a line through the origin.

4 3 2 1 0

c. Describe a proportional situation that could be modeled by this table and graph. Let x represent the number of chores you complete, and let y represent the number of hours you can spend playing video games. For every 3 chores you complete, you can play video games for 1 hour.

The graph represents a proportional relationship. The line passes through the origin.

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P R ACT I C E

29-Aug-21 4:38:30 PM

EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

5. The graph shows the number of candy bars sold and the money received.

c. Use the values from the table to justify that the relationship between the amount of money received and the number of candy bars sold is proportional.

The table shows that there is a constant unit rate, or a constant of proportionality, associated with the relationship between the amount of money received and the number of candy bars sold. The constant of proportionality is 1.5.

12 Money Received (dollars)

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

10 8 6

d. What does the point (0, 0) mean in this context?

The point (0, 0) means that if 0 candy bars are sold, the amount of money received is 0 dollars.

Number of Candy Bars Sold

a. Does the graph appear to represent a proportional relationship between the amount of money received and the number of candy bars sold? Explain how you know.

6. Logan and Shawn recorded how much money they earned working at a local restaurant over a 5-hour shift. Use the tables shown for parts (a) and (b).

The graph appears to represent a proportional relationship because the points appear to lie on a straight line that passes through the origin.

Money Logan Earned

b. Create a table of values based on the graph. Number of Candy Bars Sold, x

Money Received, y (dollars)

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Number of Hours Worked

Money Earned (dollars)

Money Shawn Earned

P R ACT I C E

Number of Hours Worked

Money Earned (dollars)

9.50

28.50

47.50

P R ACT I C E

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EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

a. Graph both sets of data in the same coordinate plane. Choose an appropriate scale and label your axes.

Remember For problems 7–10, multiply.

Logan's Earnings 60

2 3

⋅

2 5

3 4

⋅ 4

2 3

⋅

5 6

10.

4 5

⋅

5 6

Shawn's Earnings 4 15

50 Money Earned (dollars)

EUREKA MATH2

7 ▸ M1 ▸ TA ▸ Lesson 4

10 18

12 20

20 30

11. The table shows costs in dollars for different quantities of baseballs. Do the corresponding values in the table represent a proportional relationship between the cost and the number of baseballs? Explain how you know.

0.5

1.5

2.5

3.5

4.5

Hours Worked

Number of Baseballs

Cost (dollars)

5.50

12.50

The corresponding values in the table do not represent a proportional relationship between the cost and the number of baseballs. Dividing each cost by the corresponding number of baseballs does not result in a constant unit rate.

b. Describe the similarities and differences between Shawn’s and Logan’s earnings. Use the graphs to support your thinking. Both graphs show an increase in money earned with more hours worked. The graphs show that Shawn’s earnings grew at a constant rate, while Logan’s earnings did not.

12. Liam uses 3 cups of milk to make 4 batches of muffins. How many cups of milk does he need to make just 1 batch of muffins? Liam needs

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P R ACT I C E

3 4

cups of milk to make 1 batch of muffins.

P R ACT I C E

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0 0

Area of a Square, y

Cost, y (dollars)

Side Length of a Square, x

Cost, y (dollars)

100

Green Beans, x (pounds)

Number of Copies Made, x

6.30

120

300

14.70 10.50

37.50

250

12 18

0 11

7.60 3.50

3 1.5

1.10

0.5

5.15

Value of Change Collected, y (dollars)

Time, x (hours)

Bike Rental Cost, y (dollars)

Time, x (hours)

3 4

1 2

3 4 4

Water, y (cups)

Rice, x (cups)

EUREKA MATH2 7 ▸ M1 ▸ TA ▸ Lesson 4 ▸ Table Sort

This page may be reproduced for classroom use only.

20-Aug-21 4:24:18 PM

EUREKA MATH2

x 1.5 0.5 x 4.5

This page may be reproduced for classroom use only.

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1.5 0

y 0

0.5

100

150

2.5

200

3.5

300 250

0.5 0 x

0 x

1.5

2.5

3.5

7 ▸ M1 ▸ TA ▸ Lesson 4 ▸ Graph Match

6/7/2020 10:54:39 AM