# Lesson 8

Slopes Don't Have to be Positive

## 8.1: Which One Doesn’t Belong: Odd Line Out (5 minutes)

### Warm-up

This warm-up prompts students to compare four lines. It invites students to explain their reasoning and hold mathematical conversations, and allows you to hear how they use terminology and talk about lines and their properties. To allow all students to access the activity, each figure has one obvious reason it does not belong. Encourage students to move past the obvious reasons (e.g., line $$t$$ has a different color) and find reasons based on geometric properties (e.g., a slope triangle of line $$u$$ is not similar to the slope triangles of the other three lines).

So far, we have only considered lines with positive slopes. The purpose of this warm-up is to suggest similarities (same vertical and horizontal lengths of slope triangles) and differences (since they are not parallel, there is something fundamentally different going on here) between lines whose slopes have the same absolute value but opposite signs.

As students share their responses, listen for important ideas and terminology that will be helpful in the work of this lesson. Students may:

• Identify lines that have the same slope.
• Identify points where lines intersect.
• Distinguish between lines with positive and negative values for slope and be able to articulate the difference clearly.

### Launch

Arrange students in groups of 2–4 and provide access to geometry toolkits. Display the image of the four lines for all to see. Ask students to indicate when they have noticed one line that does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask each group to offer at least one reason a particular line doesn’t belong.

### Student Facing

Which line doesn’t belong?

### Activity Synthesis

After students have conferred in groups, invite each group to share one reason why a particular line might not belong. Record and display the responses for all to see. After each response, ask the rest of the class if they agree or disagree. Since there is no single correct answer to the question asking which shape does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, prompt students to use mathematical terminology (parallel, intersect, slope) correctly. Also, press students on unsubstantiated claims. For example, a student may claim that $$u$$ does not belong because it has a different slope. Ask how they know for sure that its slope is different from the other lines. Demonstrate drawing a slope triangle and computing slope.

Based on the work done up to this point in the unit, students are likely to assume that the slope of $$v$$ is $$\frac13$$. In the discussion, solicit the idea that there is something fundamentally different about line $$v$$ compared to the others. You could use informal language like “uphill” and “downhill,” or “tilt direction.” The expressions positive and negative slope do not need to be introduced at this time.

## 8.2: Stand Clear of the Closing Doors, Please (15 minutes)

### Activity

In previous activities with linear relationships, when $$x$$ increases the $$y$$ value increases as well; adding objects to a cylinder increases the water level and adding money to a bank account increases the balance. The slope of the lines that represent these relationships were positive. In this activity, students see negative slopes for the first time.

In this activity, students answer questions about a public transportation fare card context. After computing the amount left on the card after 0, 1, and 2 rides, they express regularity in repeated reasoning (MP8) to represent the amount remaining on the card after $$x$$ rides. They are told that the slope of this line is -2.5, and are prompted to explain why a negative value makes sense.

While the language is not introduced in the task statement, the value of $$x$$ for which the money on the card is 0 is called the $$x$$-intercept or horizontal intercept. Unlike the $$y$$-intercept, which can be seen in the equation $$y = 40 - 2.5x$$, the $$x$$ intercept has to be calculated: it is the value of $$x$$ for which $$0 = 40 - 2.5x$$.

### Launch

If your students are unlikely to be familiar with public transportation, you may need to give them some quick information about how a fare card works. If possible, prepare some photos related to purchasing and using a fare card. (Some example images are provided, here.)

Explain to students that someone who wants to ride the bus or subway in a city often uses a card like this. The rider pays money which a computer system associates with the card. Every time the rider wants to ride, they swipe the card, and the cost of the ride is subtracted in the computer system from the balance on the card. Eventually, the amount available on the card runs out, and the rider must spend more money to increase the amount available on the card.

Arrange students in groups of 2 and give them 5 minutes of quiet work time. Provide access to rulers. After they have discussed their responses with a partner, discuss why a negative value for the slope makes sense in the context and ways to visually tell whether a line has a positive or negative slope.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention

### Student Facing

Noah put $40 on his fare card. Every time he rides public transportation,$2.50 is subtracted from the amount available on his card.

1. How much money, in dollars, is available on his card after he takes
1. 0 rides?
2. 1 ride?
3. 2 rides?
4. $$x$$ rides?
2. Graph the relationship between amount of money on the card and number of rides.

3. How many rides can Noah take before the card runs out of money? Where do you see this number of rides on your graph?

### Launch

If your students are unlikely to be familiar with public transportation, you may need to give them some quick information about how a fare card works. If possible, prepare some photos related to purchasing and using a fare card. (Some example images are provided, here.)

Explain to students that someone who wants to ride the bus or subway in a city often uses a card like this. The rider pays money which a computer system associates with the card. Every time the rider wants to ride, they swipe the card, and the cost of the ride is subtracted in the computer system from the balance on the card. Eventually, the amount available on the card runs out, and the rider must spend more money to increase the amount available on the card.

Arrange students in groups of 2 and give them 5 minutes of quiet work time. Provide access to rulers. After they have discussed their responses with a partner, discuss why a negative value for the slope makes sense in the context and ways to visually tell whether a line has a positive or negative slope.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention

### Student Facing

Noah put $40 on his fare card. Every time he rides public transportation,$2.50 is subtracted from the amount available on his card.

1. How much money, in dollars, is available on his card after he takes

1. 0 rides?
2. 1 ride?
3. 2 rides?
4. $$x$$ rides?
2. Graph the relationship between amount of money on the card and number of rides.

3. How many rides can Noah take before the card runs out of money? Where do you see this number of rides on your graph?

### Launch

Show the image and ask students “What do you notice? What do you wonder?” Expect students to notice that the line is horizontal (or the amount of money on the card does not change). They may wonder why the line is horizontal or what its slope is.

Keep students in the same groups. 5 minutes of quiet think time, followed by partner and class discussion. Before students begin the task, ensure that they understand that the $$x$$-axis no longer represents number of rides, but rather, different days in July.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to record what they notice and wonder prior to being expected to share these ideas with others.
Supports accessibility for: Language; Organization
Writing, Speaking: MLR5 Co-Craft Questions. Present the graph that shows the amount on Han’s fare card for every day of last July, without revealing the questions that follow. Give students time to write down possible mathematical questions that can be asked about the situation, and then invite them to share their questions with a partner. Listen for and amplify questions that wonder about the slope of a horizontal line. This will help students make sense of a graph with a “flat line,” or a line having a zero slope, by generating questions using mathematical language.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

### Student Facing

Here is a graph that shows the amount on Han’s fare card for every day of last July.

1. Describe what happened with the amount on Han’s fare card in July.

2. Plot and label 3 different points on the line.

3. Write an equation that represents the amount on the card in July, $$y$$, after $$x$$ days.

4. What value makes sense for the slope of the line that represents the amounts on Han’s fare card in July?

### Launch

Show the image and ask students “What do you notice? What do you wonder?” Expect students to notice that the line is horizontal (or the amount of money on the card does not change). They may wonder why the line is horizontal or what its slope is.

Keep students in the same groups. 5 minutes of quiet think time, followed by partner and class discussion. Before students begin the task, ensure that they understand that the $$x$$-axis no longer represents number of rides, but rather, different days in July.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to record what they notice and wonder prior to being expected to share these ideas with others.
Supports accessibility for: Language; Organization
Writing, Speaking: MLR5 Co-Craft Questions. Present the graph that shows the amount on Han’s fare card for every day of last July, without revealing the questions that follow. Give students time to write down possible mathematical questions that can be asked about the situation, and then invite them to share their questions with a partner. Listen for and amplify questions that wonder about the slope of a horizontal line. This will help students make sense of a graph with a “flat line,” or a line having a zero slope, by generating questions using mathematical language.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

### Student Facing

Here is a graph that shows the amount on Han’s fare card for every day of last July.

1. Describe what happened with the amount on Han’s fare card in July.
2. Plot and label 3 different points on the line.
3. Write an equation that represents the amount on the card in July, $$y$$, after $$x$$ days.

4. What value makes sense for the slope of the line that represents the amounts on Han’s fare card in July?

### Student Facing

#### Are you ready for more?

Let’s say you have taken out a loan and are paying it back. Which of the following graphs have positive slope and which have negative slope?

1. Amount paid on the vertical axis and time since payments started on the horizontal axis.
2. Amount owed on the vertical axis and time remaining until the loan is paid off on the horizontal axis.
3. Amount paid on the vertical axis and time remaining until the loan is paid off on the horizontal axis.

### Activity Synthesis

Display the graph for all to see, and ask students to articulate what they think the slope of the graph is, and why. The goal is to understand that a slope of 0 makes sense because no money is added or subtracted each day. If students have been thinking in terms of uphill and downhill lines, they might describe this line as “flat,” indicating that the slope can’t be positive or negative, so 0 makes sense. Thinking of slope as the quotient of horizontal displacement by vertical displacement for two points on a line is very effective here: the vertical displacement is 0 for any two points on this line, and so the quotient or slope is also 0.

If no one brings it up, ask students what would happen if we tried to create a slope triangle for this line. They might claim that it would be impossible, but suggest that we can think of a slope “triangle” where the vertical segment has length 0. In other words, it is possible to imagine a horizontal line segment as a triangle whose base is that segment and whose height is 0. If possible, display and demonstrate with the following: https://ggbm.at/vvQPutaJ.

## 8.4: Payback Plan (10 minutes)

### Optional activity

This activity gives students an opportunity to interpret the graph of a line in context, including the meaning of a negative slope and the meaning of the horizontal and vertical intercepts.

### Launch

3 minutes of quiet work time, followed by 2 minutes to confer in small groups to verify answers.

Reading: MLR6 Three Reads. Use this routine for Elena’s situation to support reading comprehension for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., Elena is paying back her brother every week.). In the second read, ask students to look for quantities represented in the graph (e.g., at week 0, Elena owes him $18; at 6 weeks, she has paid him back completely). In the third read, ask students to brainstorm possible strategies to answer the question: “What is the slope of the line and what does it represent?” This will help students reflect on the situation and interpret the slope in the given context. Design Principle(s): Support sense-making; Maximize meta-awareness ### Student Facing Elena borrowed some money from her brother. She pays him back by giving him the same amount every week. The graph shows how much she owes after each week. Answer and explain your reasoning for each question. 1. What is the slope of the line? 2. Explain how you know whether the slope is positive or negative. 3. What does the slope represent in this situation? 4. How much did Elena borrow? 5. How much time will it take for Elena to pay back all the money she borrowed? ### Student Response Teachers with a valid work email address can click here to register or sign in for free access to Student Response. ### Anticipated Misconceptions Students may interpret money owed as negative. In this setup, the axis is labeled so that money owed is treated as positive. ### Activity Synthesis Ask students to share answers to each question and indicate how to use the graph to find the answers. For example, draw a slope triangle for the first question: the slope is -3 rather than 3, because the number of dollars owed is decreasing over time. Label the vertical intercept for the amount Elena borrowed and the horizontal intercept for the time it took her to pay back the loan. ## Lesson Synthesis ### Lesson Synthesis In this lesson, students learned that the slope of a line can be a negative value or 0. They saw some linear relationships with a negative slope and some with 0 slope. Students learned about cues to identify whether a graphed line has a positive slope, a negative slope, or 0 slope. Display the graph for all to see. Ask students to pretend that their partner has been absent from class for a few days. Their job is to explain, verbally or in writing, how someone would figure out the slope of one of the graphed lines. Then, switch roles and listen to their partner explain how to figure out the slope of the other line. ## 8.5: Cool-down - The Slopes of Graphs (5 minutes) ### Cool-Down Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs. ## Student Lesson Summary ### Student Facing At the end of winter in Maine, the snow on the ground was 30 inches deep. Then there was a particularly warm day and the snow melted at the rate of 1 inch per hour. The graph shows the relationship between the time since the snow started to melt and the depth of the snow. The slope of the graph is -1 since the rate of change is -1 inch per hour. That is, the depth goes down 1 inch per hour. The vertical intercept is 30 since the snow was 30 inches deep when the warmth started to melt the snow. The two slope triangles show how the rate of change is constant. It just also happens to be negative in this case since after each hour that passes, there is 1 inch less snow. Graphs with negative slope often describe situations where some quantity is decreasing over time, like the depth of snow on warm days or the amount of money on a fare card being used to take rides on buses. Slopes can be positive, negative, or even zero! A slope of 0 means there is no change in the $$y$$-value even though the $$x$$-value may be changing. For example, Elena won a contest where the prize was a special pass that gives her free bus rides for a year. Her fare card had \$5 on it when she won the prize. Here is a graph of the amount of money on her fare card after winning the prize:

The vertical intercept is 5, since the graph starts when she has \\$5 on her fare card. The slope of the graph is $$0$$ since she doesn’t use her fare card for the next year, meaning the amount on her fare card doesn’t change for a year. In fact, all graphs of linear relationships with slopes equal to 0 are horizontal—a rate of change of 0 means that, from one point to the next, the $$y$$-values remain the same.