This Warm-up prompts students to compare four graphs. 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 items 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 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.
Activity
None
Which three go together? Why do they go together?
A
B
C
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 “straight line,” “solid line,” “steep,” “shallow,” and “origin,” 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?”
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.
9.2
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2.c
Represent proportional relationships by equations. For example, if total cost is proportional to the number of items purchased at a constant price , the relationship between the total cost and the number of items can be expressed as .
In this activity students identify correspondences between different representations of a proportional relationship, including a table, a graph, and an equation. Students are guided to notice:
Any pair of positive values determine a proportional relationship.
Given a point other than the origin on the graph of a line through the origin, the constant of proportionality is always (for all points other than the origin).
In an equation that represents the relationship, the constant of proportionality appears as the coefficient of .
The constant of proportionality is the -coordinate when is 1, that is, is a point on the graph.
As students recognize these correspondences between representations, they are making use of structure (MP7).
There are three different starting points given, and each student is assigned to work with one of them. After creating and analyzing their representations, students compare their work with others who had a different starting point. Monitor for students who notice these connections across representations while working with each of the three different given points .
In the digital version of the activity, students use an applet to manipulate the graph of a proportional relationship. The applet allows students to see how the equation and the values in the table change when the point is moved. This activity works best when each student has access to the applet, because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the applet would be helpful during the Activity Synthesis.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Activity Synthesis
The goal of this discussion is to highlight connections between different representations of a proportional relationship. Display an example of the table, equation, and graph for each of the three starting points for all to see.
Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:
“How do these different representations show the same information?”
“Are there any benefits or drawbacks to one representation compared to another?”
“How does the constant of proportionality show up in each representation?”
Ensure that all the important connections are highlighted:
A graph of a line through the origin and passing through the first quadrant represents a proportional relationship.
The value of computed from any point on that line (other than the origin) is the constant of proportionality.
An equation of the relationship is given by where is for any point on the graph other than the origin.
9.3
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2.b
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Represent proportional relationships by equations. For example, if total cost is proportional to the number of items purchased at a constant price , the relationship between the total cost and the number of items can be expressed as .
The purpose of this activity is to help students see that a proportional relationship between two quantities can be represented by two different graphs, depending on which quantity is graphed on which axis. This builds on previous activities where students saw that a proportional relationship is associated with two different, reciprocal rates. In this situation, the two rates are the number of balloon animals per minute and the number of minutes per balloon animal. Students attend to precision (MP6) as they specify the meaning of each constant of proportionality.
In the digital version of the activity, students use an applet to graph and compare proportional relationships on two sets of axes. The applet allows students to add, remove, adjust, and label points and lines. The digital version may help students graph quickly and accurately so they can focus more on the mathematical analysis.
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
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.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, use a different color for each person to highlight the connection between the graph, equation, and constant of proportionality. Supports accessibility for: Visual-Spatial Processing
Activity
None
Andre and Jada had a contest making balloon animals.
Andre made 10 balloon animals in 3 minutes.
Jada made 12 balloon animals in 5 minutes.
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 to represent time in minutes and to represent the number of balloon animals.
Andre:
Jada:
For each equation, what does the constant of proportionality tell you?
Repeat the previous steps for the second graph.
Andre:
Jada:
Activity Synthesis
First, invite a few students to share their reasoning for how they determined which points represented Andre’s work and which represented Jada’s. Ensure that students are in agreement before continuing.
Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the question about Andre’s rate by correcting errors, clarifying meaning, and adding details.
Display this first draft:
“Andre was making balloon animals at a rate of .”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
The key takeaways are:
There are multiple ways to express the constant of proportionality between two quantities. The important thing is that we communicate the meaning of the number clearly.
, , , and 0.3 are all possible correct answers:
or represents the number of balloon animals per minute that Andre made.
or 0.3 represents the number of minutes per balloon animal that Andre made.
and are reciprocals of each other.
Lesson Synthesis
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.”
“What is the constant of proportionality for the relationship shown on the first graph? How do you know?” (2. The -value is 2 when the -value is 1. You can also divide 6 by 3 to get 2.)
“What equation could represent this relationship?” ( or equivalent)
“What is the constant of proportionality for the relationship shown on the second graph? How do you know?” (. You can see that the -value is when the -value is 1. You can also divide 3 by 6 to get 0.5.)
“What equation could represent this relationship?” ( or equivalent)
Student Lesson Summary
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.
We could say that the elapsed time is proportional to the volume of water. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of of a minute per milliliter.
We could say that the volume of water is proportional to the elapsed time. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of 5 milliliters per minute.
Let’s use to represent volume in milliliters and to represent time in minutes. Here are graphs and equations that represent both ways of thinking about this relationship:
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 as the independent variable, and the graph on the right has as the independent variable.
Standards Alignment
Building On
Addressing
7.RP.A.2.d
Explain what a point on the graph of a proportional relationship means in terms of the situation, with special attention to the points and where is the unit rate.
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 -values for the -values 2 and 6 and then continue answering the questions.
Select work from students with different starting points to share later.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, For example, present one question at a time and monitor students to ensure they are making progress throughout the activity. Supports accessibility for: Organization, Attention
Activity
None
Your teacher will assign you one of these three points:
, , .
On the coordinate plane, plot and label only your assigned point.
Graph the proportional relationship that is defined by your point. That is, use a ruler to draw a line that starts at the origin, goes through your point, and continues to the edge of the grid.
Use your graph to find the -value that goes with each of these -values.
2
6
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 for each row. What do you notice about these values?
Write an equation that represents the relationship between and .
What is the -coordinate of your graph when the -coordinate is 1? Plot and label this point on your graph.
Based on your observations, describe any connections you see between the graph, the table, and the equation.
Compare your representations with the rest of your group. Discuss what is the same and what is different about:
Your graphs.
Your tables.
Your equations.
Student Response
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Building on Student Thinking
Student Response
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Building on Student Thinking
7.RP.A.2.d
Explain what a point on the graph of a proportional relationship means in terms of the situation, with special attention to the points and where is the unit rate.
Explain what a point on the graph of a proportional relationship means in terms of the situation, with special attention to the points and where is the unit rate.
