Graphs of Proportional Relationships
5 min
Teacher Prep
Setup
Narrative
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 Task
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.
Sample Response
Activity Synthesis (Teacher Notes)
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.)
Standards
- 7.RP.2·Recognize and represent proportional relationships between quantities.
- 7.RP.A.2·Recognize and represent proportional relationships between quantities.
- 8.EE.5·Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <em>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</em>
- 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. <span>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</span>
10 min
20 min
Knowledge Components
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Graphs of Proportional Relationships
5 min
Teacher Prep
Setup
Narrative
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 Task
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.
Sample Response
Activity Synthesis (Teacher Notes)
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.)
Standards
- 7.RP.2·Recognize and represent proportional relationships between quantities.
- 7.RP.A.2·Recognize and represent proportional relationships between quantities.
- 8.EE.5·Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <em>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</em>
- 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. <span>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</span>