In this Warm-up, students work with the same proportional relationship shown on two sets of axes that are scaled differently. The purpose is to make explicit that the same proportional relationship can appear to have different steepness depending on the axes.
Launch
Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Student Lesson in Spanish
Activity
None
Student Task Statement
Here are two graphs that could represent a variety of different situations.
Andre claims that the line in the graph on the left has a greater slope because it is steeper. Do you agree with Andre? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to emphasize the importance of paying attention to scale when making sense of graphs. Display the two images from the activity for all to see. Identify 1–2 students to share their reasoning. Here are some questions for discussion:
“How can one graph look steeper yet still have the same slope as another graph?” (The two graphs are drawn using different scales, making them look different even though the value of their slopes is equivalent.)
“Are the two slope triangles shown similar?” (Yes. The slope triangle on the left can be dilated and translated to match the slope triangle on the right, making the two triangles similar.)
“What would happen if these 2 lines were graphed on the same set of axes?” (They would overlap and look like the same line.)
2.2
Activity
10 mins
Card Sort: Proportional Relationships
Standards Alignment
Building On
Addressing
8.EE.B
Understand the connections between proportional relationships, lines, and linear equations.
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
The purpose of this activity is for students to identify the same proportional relationship graphed using different scales. Students will first sort the graphs based on what proportional relationship they represent and then write an equation representing each relationship. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7).
Monitor for and select groups using different strategies to match graphs to share later. For example, some groups may identify the unit rate for each graph in order to match while others may choose to write equations first and use those to match their graphs.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 4 and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
Give students 1 minute to place all the cards face up and start thinking about possible ways to sort the cards into categories.
Pause the class and select 1–3 students to share the categories they identified.
Discuss as many different categories as time allows.
Attend to the language that students use to describe their categories and graphs, giving them opportunities to describe their graphs more precisely. Highlight the use of terms like “slope,” “scale,” and “constant of proportionality.” If necessary, remind students that the constant of proportionality is the number that values for one quantity are each multiplied by to get the values for the other quantity. After a brief discussion, invite students to open their books or devices and continue with the activity.
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed , so I matched . . . .” Encourage students to challenge each other when they disagree. Advances: Speaking, Conversing
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory
Activity Synthesis
The goal of this discussion is for students to understand that looking only at the steepness of the line without paying attention to the numbers on the axes can hide the actual relationship between the two variables.
Once all groups have completed the Card Sort, ask previously selected groups to share their strategies for grouping the graphs. Discuss the following:
“Which graphs were tricky to group? Explain why.”
“Did you need to make adjustments in your groups? What might have caused an error? What adjustments were made?”
2.3
Activity
20 mins
Different Scales
Standards Alignment
Building On
Addressing
8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
In this activity, students graph a proportional relationship on two differently scaled axes and compare the proportional relationship to an already-graphed nonproportional relationship on the same axes. Students make sense of the intersections of the two graphs by reasoning about the situation and consider which scale is most helpful: zoomed in, or zoomed out. In this case, which graph is most helpful depends on the questions asked about the situation (MP4).
This is the first time Three Reads (Math Language Routine 6) 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 important quantities. During the third read, the final prompt is revealed and students brainstorm possible strategies to answer the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing to a solution. 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.
Launch
Arrange students in groups of 2. Provide access to straightedges.
Use MLR6 Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the graphs, without revealing the questions.
In the first read, students read the problem with the goal of comprehending the situation.
Read the problem aloud while everyone else reads along, and then ask, “What is this situation about?” Allow 1 minute to discuss with a partner and then share with the whole class. A typical response may be, “Two large water tanks are filling with water. One of them is filling at a constant rate, while the other is not. Both graphs represent Tank A. Tank B only has an equation.” 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 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 “liters of water in Tank A; liters of water in Tank B; amount of time that has passed in minutes; constant rate of 1/2 liters per minute.”
In the third read, students brainstorm possible solution strategies to answer 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 on sketching and labeling a graph for Tank B on each of the axes and ask, “What are some ways one might solve this?” Instruct students to think of ways to approach the questions without actually solving. Give students 1 minute of quiet think time followed by another minute to discuss with their partner. Invite students to name some possible strategies referencing quantities from the second read. Provide these sentence frames as partners discuss: “To draw a graph for Tank B, I would . . . .” “One way to approach the question about finding the time when the tanks have the same amount of water would be to . . . .” and “I would use the first/second graph to find . . . .”
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 the ideas for each question.
Give students time to complete the rest of the activity followed by a whole class discussion.
Activity
None
Activity Synthesis
The goal of this discussion is to emphasize how selecting an appropriate scale is important when creating a graph from scratch. Begin the discussion by asking students:
“What information can be seen using the second graph that can’t be seen with the first?” (Tank A starts out with more water than Tank B at first, and then with less water than Tank B later on. The two tanks have the same amount of water after 60 minutes.)
“If someone only looked at the first graph, what might they think (incorrectly)?” (The first graph makes it look like Tank A will always have more water than Tank B.)
“In what situation might the first graph be more useful?” (If someone wanted to answer a question about the tanks after only 1 or 2 minutes.)
Explain that if making a graph from scratch, it is important to first check what questions are being asked. Some things to consider are:
Should both axes have the same scale?
How large are the numbers in the problem? Does each axis need to extend to 10 or 100?
What will you count by? 1s? 5s? 10s?
Lesson Synthesis
Display this blank coordinate plane for all to see and provide pairs of students with graph paper.
Ask pairs to draw a copy of the coordinate plane and give a signal when they have finished. (Students may need to be warned to leave room on their graph paper for a second graph as sometimes students like to draw graphs that fill all the space they are given.) Invite a student to propose a proportional relationship that they consider to have a “steep” line for the class to graph on the axes.
For example, say a student proposes . Have students complete their graph in pairs, then add the line representing the equation to the graph on display. Next, ask students to make a second graph with the same horizontal scale, but with a vertical scale that makes not look as steep when graphed. After students have made the new graph, invite students to share and explain how they decided on their new vertical scale.
Conclude by reminding students that all these graphs of are correct since they all show a proportional relationship with a constant of proportionality equal to 6. Ask students, “Can you think of a reason someone might want to graph this relationship with such a large vertical scale?” (If they needed to also graph something like , they would need a pretty big vertical scale in order to see both lines.)
Student Lesson Summary
The scales we choose when graphing a relationship often depend on what information we want to know. For example, consider two water tanks filled at different constant rates.
The relationship between time in minutes and volume in liters of Tank A can be described by the equation .
For Tank B the relationship can be described by the equation
These equations tell us that Tank A is being filled at a constant rate of 2.2 liters per minute and Tank B is being filled at a constant rate of 2.75 liters per minute.
If we want to use graphs to see at what times the two tanks will have 110 liters of water, then using an axis scale from 0 to 10, as shown here, isn't very helpful.
graph, horizontal axis, time in minutes, scale 0 to 9, by 1's. vertical axis, volume in liters, 0 to 9, by 1's. lines, first line passing through origin and 2 comma 5 and 5 tenths. second line passing through origin and 2 comma 4 and 5 tenths.
graph, horizontal axis, time in minutes, scale 0 to 90, by 10's. vertical axis, volume in liters, 0 to 140, by 10's. 2 lines, first line passing through origin and 40 comma 110. second line passing through origin and 50 comma 110.
If we use a vertical scale that goes to 150 liters, a bit beyond the 110 we are looking for, and a horizontal scale that goes to 100 minutes, we get a much more useful set of axes for answering our question.
Now we can see that the two tanks will reach 110 liters 10 minutes apart—Tank B after 40 minutes of filling and Tank A after 50 minutes of filling.
It is important to note that both of these graphs are correct, but one uses a range of values that helps answer the question. In order to always pick a helpful scale, we should consider the situation and the questions asked about it.
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Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Your teacher will give you a set of cards. Each card contains a graph of a proportional relationship.
Sort the graphs into groups based on what proportional relationship they represent.
Write an equation for each different proportional relationship you find.
Student Response
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Building on Student Thinking
Student Task Statement
Two large water tanks are filling with water. Tank A is not filled at a constant rate, and the relationship between its volume of water and time is graphed on each set of axes. Tank B is filled at a constant rate of liters per minute. The relationship between its volume of water and time can be described by the equation , where is the time in minutes, and is the total volume in liters of water in the tank.
graph, horizontal axis, time in minutes, scale 0 to 1 and 8 tenths, by 2 tenth's. vertical axis, volume in liters, 0 to 1 and 8 tenths, by 2 tenth's. curved line passing through origin, 1 comma 1 and 1 and 4 tenths comma 1 and 3 tenths.
graph, horizontal axis, time in minutes, scale 0 to 80, by 20's. vertical axis, volume in liters, 0 to 50, by 10's. curved line passing through origin, 20 comma 12 and 60 comma 30.</p>
Sketch and label a graph of the relationship between the volume of water and time for Tank B on each of the coordinate planes.
Answer the following questions and say which graph you used to find your answer.
After 30 seconds, which tank has the most water?
At approximately what times do both tanks have the same amount of water?
At approximately what times do both tanks contain 1 liter of water? 20 liters?
