This Warm-up prompts students to compare four lines. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the lines in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three lines that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Student Lesson in Spanish
Activity
None
Which three go together? Why do they go together?
4 lines plotted on a grid, lines labeled a,b,c,d. line a, positive slope, intersects line d. line d, negative slope, intersects lines a, b, c. line b, positive slope, intersects lines d and c. line c, positive slope, intersects lines b and 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 go 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 the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “parallel,” “intersect,” and “slope triangle,” 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?”
9.2
Activity
Standards Alignment
Building On
Addressing
8.EE.B
Understand the connections between proportional relationships, lines, and linear equations.
In this activity, students see negative slopes for the first time as they answer questions about a public transportation fare card. After computing the amount left on the card after 0, 1, and 2 rides, they express regularity in repeated reasoning to represent the amount remaining on the card after rides (MP8). Students reason about why a negative value for the slope of this line makes sense (MP2).
While the language is not introduced in the task statement, the value of for which the money on the card is 0 is called the -intercept or horizontal intercept. Unlike the -intercept, which can be seen in the equation , the -intercept has to be calculated. It is the value of for which .
In the digital version of the activity, students use an applet to graph the relationship between the number of rides Noah has taken on public transportation and the amount of money left on his fare card. The applet allows students to graph this relationship quickly and precisely. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.
Activity Synthesis
The purpose of this discussion is for students to understand what a negative slope means in context. Begin by asking students, "Why does it make sense to say the slope of this graph is -2.5 rather than 2.5?" Ensure that students can articulate some version of “For every ride, the amount on the card decreases by $2.50, and this is why it makes sense that the slope is -2.5.”
Invite students to share their answers to the question about when Noah will run out of money on his card. (After 16 rides, the card has no money left, which can be seen at the point .) Explain that just like the point on the graph is called the vertical intercept, the point is called the horizontal intercept. In this situation, the vertical intercept tells how much money Noah started with on the card, and the horizontal intercept tells how many rides Noah can take before the card has no more money on it.
Display this image of lines and from the Warm-up and draw in two slope triangles as shown.
Ask students what they notice about the vertical and horizontal lengths of the two slope triangles. Explain that even though they both have a vertical length of 1 and horizontal length of 3, they don’t have the same slope. Ask students which line has a slope of and which line has a slope of . Validate students’ use of informal language to describe the differences between the two lines. For example, students may say that if you put a pencil on a line and move it along the line from left to right, line goes “uphill” but line goes “downhill.”
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to “Why does it make sense to say the slope of this graph is -2.5 rather than 2.5?” Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
9.3
Activity
Standards Alignment
Building On
Addressing
8.EE.B
Understand the connections between proportional relationships, lines, and linear equations.
This activity introduces the idea of a slope of zero by considering a fare card where the amount on the card does not go up or down at all.
Launch
Tell students to close their books or devices (or to keep them closed). Display the graph for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the graph.
If the fact that the -axis represents different days in July rather than the number of rides does not come up during the conversation, ask students to discuss this idea.
Tell students to open their books or devices and give them 3–4 minutes to complete the task, followed by a partner then whole-class discussion.
Representation: Internalize Comprehension. Provide students with a graphic, such as a two-column table, to record what they notice and wonder prior to being expected to share these ideas with others. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
This graph shows the amount of money in dollars that is on Han’s fare card for every day of last July.
Describe what happened with the money on Han’s fare card in July.
Plot and label 3 different points on the line.
Write an equation where represents the day in July and represents the dollars on the card.
What value makes sense for the slope of the line that represents the money on Han’s fare card in July?
Student Response
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Building on Student Thinking
If students have trouble writing an equation to represent the situation, consider asking:
“What do you notice about the 3 points on the line that you plotted and labeled?”
“How can you figure out the -value if you know the -value?”
Activity Synthesis
The goal of this discussion is for students to understand a context where a slope of 0 makes sense. Display the graph from the task for all to see. Ask students what they think the slope of the graph is and to explain their reasoning. Ensure that students 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.
If not mentioned by students, ask what would happen if they tried to create a slope triangle for this line. They may claim that it would be impossible, but suggest thinking of a slope “triangle” where the vertical segment has length 0. 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 possible, display and demonstrate this with the following
: https://ggbm.at/vvQPutaJ.
Lesson Synthesis
The goal of this discussion is for students to determine cues that identify whether a graphed line has a positive slope, a negative slope, or zero slope. Display the graph for all to see.
Create a classroom display that is divided into 3 sections and invite students to explain how they know if a line has a positive, negative, or zero slope. Record their responses on chart paper, and label the image when appropriate. Important ideas to record include:
As the value increases, the value increases (positive slope), decreases (negative slope), or stays the same (zero slope).
From left to right, the graph of the line looks like it is going uphill (positive slope), downhill (negative slope), or flat (zero slope).
If time allows, give students an opportunity to practice thinking about contexts with positive, negative, and zero slopes. Display a blank set of coordinate axes and give students a label for the horizontal axis, such as time in hours. Ask students to think of labels for the vertical axis that might result in lines with positive, negative, or zero slopes. For example, graphing the total miles driven during a road trip might have a positive slope, while graphing miles left until reaching the destination might have a negative slope.
Student Lesson Summary
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 remaining snow.
Graphs with a negative slope often describe situations where some quantity is decreasing over time.
Since the depth of the snow decreases by 1 inch per hour, the rate of change is -1 inch per hour and the slope of this graph is -1. The vertical intercept is 30 since the snow was 30 inches high before it started to melt.
graph on grid, origin O. horizontal axis, time since snow started to melt in hours, scale 0 to 11, by 1's. vertical axis, depth of snow in inches, scale 0 to 30, by 5's. line with negative slope drawn that goes through points 2 comma 28, 3 comma 27. triangle with down 1 right 1 drawn below these 2 points. line also goes through 5 comma 25 and 10 comma 20. triangle with down 5 right 5 drawn below these 2 points.
Graphs with a slope of 0 describe situations where there is no change in the -value even though the -value is changing.
graph on a grid, origin O. horizontal axis, time since winning contest in days, scale 0 to 11, by 1's. vertical axis, balance on fare card in dollars, 0 to 6, by 1's. horizontal line through 2 days comma 5 dollars and 5 days comma 5 dollars.
For example, Elena wins a prize that gives her free bus rides for a year. Her fare card already 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. Since she doesn’t need to add or use money from her fare card for the next year, the amount on her fare card will not change. The rate of change is 0 dollars per day and the slope of this graph is 0. All graphs of linear relationships with slopes of 0 are horizontal.
Standards Alignment
Building On
Addressing
8.EE.B
Understand the connections between proportional relationships, lines, and linear equations.
If your students are unlikely to be familiar with public transportation, provide some 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.)
Explain to students that someone who wants to ride a bus or train often uses a card like this. The rider adds money to the card and the cost of each ride they take is subtracted from the balance on the card. Eventually, the money on the card runs out, and more must be added before taking additional rides.
Arrange students in groups of 2. Provide access to straightedges. Give students 4–5 minutes of quiet work time, followed by a partner then whole-class discussion.
Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies, such as a device that can run the digital applet. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization
Activity
None
Noah has \$40 on his fare card. Every time he rides public transportation, \$2.50 is subtracted from the amount available on his card.
How much money, in dollars, is available on his card after he takes
0 rides?
1 ride?
2 rides?
rides?
How many rides can Noah take before the card runs out of money? Where would you see this number of rides on a graph?
Graph the relationship between amount of money on the card and number of rides.
