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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 graphs 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.
Which three go together? Why do they go together?
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 “straight line,” “solid line,” “steep,” “shallow,” and “origin,” and to clarify their reasoning as needed. Consider asking:
If time allows, invite 2–3 students to briefly share what they notice all of the figures have in common (for example, they are all graphs in the first quadrant, they all go up from left to right, the axes are labeled with numbers but not with quantities). The purpose of this concluding share out is to provide more opportunities for students to use terminology to describe aspects of graphs.
Share with students, “Today we looked at different representations of the same proportional relationship. We found connections between the table, graph, and equation. We also looked at a pair of graphs that showed the same proportional relationship but swapped which quantity was on each axis.”
If desired, use this example to review these concepts.
“A person uses balloons and cardboard tubes to make toy barbells.”
Imagine that a faucet is leaking at a constant rate and that every 2 minutes, 10 milliliters of water leaks from the faucet. There is a proportional relationship between the volume of water and elapsed time.
Let’s use
A line graph has a horizontal v-axis and a vertical t-axis. The line begins at the origin and moves upwards and to the right passing through the points with coordinates (1 comma 1/5) and (10 comma 2). The graph is labeled with the equation t = 1/5 v.
A line graph has a horizontal t-axis and vertical v-axis. The line begins at the origin and moves upwards and to the right passing through the points with coordinates (1 comma 5) and (2 comma 10). The graph is labeled with the equation v =5 t.
Even though the relationship between time and volume is the same, we are making a different choice in each case about which variable to view as the independent variable. The graph on the left has
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Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers. Give students 5–7 minutes of quiet work time followed by small group discussion.
As students complete the third question, give them the table that goes with their assigned point, from the blackline master. They should use the table to check their
Select work from students with different starting points to share later.
Your teacher will assign you one of these three points:
Use your graph to find the
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Your teacher will give you a completed table. Use it to check your values.
Choose three rows, other than the row that represents the origin, from the completed table. Record the values and compute
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What is the
Compare your representations with the rest of your group. Discuss what is the same and what is different about:
Keep students in the same groups. Consider displaying an image of balloon animals or a video of someone making a balloon animal to familiarize students with the context.
Andre and Jada had a contest making balloon animals.
Here are two different graphs that both represent this situation.
A graph on a coordinate plane. The graph has two points. The horizontal axis is labeled "time in minutes" and the vertical axis is labeled "number of balloon animals." The first point is to the right of the vertical axis and high above the horizontal axis. The second point is to the right and above the first point.
A graph on a coordinate plane. The graph has two points. The horizontal axis is labeled "number of balloon animals" and the vertical axis is labeled "time in minutes." The first point is far to the right of the vertical axis and above the horizontal axis. The second point is to the right and above the first point.
On the first graph, which point shows Andre’s work and which shows Jada’s work? Label them.
Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point.
Write an equation for each line. Use
Andre:
Jada:
For each equation, what does the constant of proportionality tell you?
Repeat the previous steps for the second graph.
Andre:
Jada: