Lesson 2

Representing Proportional Relationships

2.1: An Unknown Situation (5 minutes)

Warm-up

In the previous warm-up, students compared different proportional relationships on two sets of axes that were scaled the same. In this warm-up, students will work with two sets of axes scaled differently and the same proportional relationship. The purpose of this warm-up is to make explicit that the same proportional relationship can appear to have different steepness depending on the axes, which is why paying attention to scale is important when making sense of graphs or making graphs from scratch. Students learned how to graph and write equations for proportional relationships in previous grades, so this warm-up also helps refresh those skills.

Identify students using different strategies to graph the relationship on the new axes. For example, since this is a proportional relationship, some students may scale up the point \((8,14)\) to something like \((40,70)\) and then plot that point before drawing a line through it and the point \((0,0)\). Other students may use the equation they wrote and one of the \(x\)-values marked on the new axes to find a point on the line and draw in the line from there.

Launch

Give 2–3 minutes of quite work time followed by a whole-class discussion.

Student Facing

Here is a graph that could represent a variety of different situations.

graph. horizontal axis, scale 0 to 22, by 2's. vertical axis, scale 0 to 28, by 2's. line passing through origin and 8 comma 14. 
  1. Write an equation for the graph.
  2. Sketch a new graph of this relationship.

    graph, horizontal axis, scale 0 to 120, by 20's. vertical axis, 0 to 210, by 35's.

Student Response

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Activity Synthesis

Display the two images from the activity for all to see. Invite previously identified students to share how they graphed the relationship on the new line.

Ask students, “Which graph looks steeper to you?” Students may see the first line as steeper, even though the two lines have the same slope. It is important students understand that this is one of the reasons mathematics uses numbers (slope) to talk about the steepness of lines and not “looks like.”

2.2: Card Sort: Proportional Relationships (15 minutes)

Activity

The purpose of this activity is for students to identify the same proportional relationship graphed using different scales. Students will first sort the cards based on what proportional relationship they represent and then write an equation representing each relationship. Identify and select groups using different strategies to match graphs to share during the Activity Synthesis. 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.

Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity if students need additional practice identifying features of the graphs of proportional relationships, such as the unit rate, or writing equations for a proportional relationship given the graph.

Launch

Arrange students in groups of 4. Provide each group with a set of 12 pre-cut slips.

Speaking: MLR8 Discussion Supports. To support students to produce statements about proportional relationships, provide sentence frames for students to use when they describe the reasoning for their matches. For example, “ ____ and ____ match/don’t match, because ____.” Encourage use of relevant vocabulary such as “constant of proportionality” and “unit rate.”
Design Principle(s): Support sense-making; Optimize output (for explanation)

Student Facing

Your teacher will give you 12 graphs of proportional relationships.

  1. Sort the graphs into groups based on what proportional relationship they represent.
  2. Write an equation for each different proportional relationship you find.

Student Response

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Anticipated Misconceptions

If students have trouble recalling the meaning of the constant of proportionality, remind them that one way to think about it is “the change in \(y\) for every unit change in \(x\).” 

Activity Synthesis

As a result of this conversation, students should understand that the scale of the axes a graph is drawn on can hide the actual relationship between the two variables if you just look at the steepness of the line without paying attention to the numbers on the axes.

Ask previously selected groups to share their strategies for matching the graphs. Highlight uses of relevant vocabulary such as “constant of proportionality” or “unit rate.”

Have students look at Card A. Ask students, “Do you think this graph looks like \(y=\frac14x\)? Why or why not?” Possible reasons from students are:

  • No, I think this graph looks like \(y=x\).
  • I would expect \(y=\frac14x\) to be less steep since an increase in \(x\) means \(\frac14\) as much of an increase in \(y\).
  • Since the \(x\)-scale is four times the \(y\)-scale, the line looks steeper that I expected.

Give students a moment to identify another card that they think does not “look like” the equation and select a few students to share the graph they chose and explain their thinking.

Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention

2.3: Different Scales (15 minutes)

Activity

Building off the work in the previous activity, students now graph a proportional relationship on two differently scaled axes and compare the proportional relationship to an already-graphed non-proportional relationship on the same axes. Students are asked to make sense of the intersections of the two graphs by reasoning about the situation and consider which scale is most helpful: the zoomed in, or the zoomed out. In this case, which graph is most helpful depends on the questions asked about the situation.

Teacher Notes for IM 6–8 Math Accelerated

The support for English language learners in this activity states that “This is the first time Math Language Routine 6 is suggested as a support in this course,” and follows with a detailed description of how the routine is meant to be used in the activity. While the detailed descriptions are still accurate, due to how lessons and units are arranged in IM 6–8 Math Accelerated, this instance is no longer the first time a specific math language routine is seen.

The activity synthesis references choices students will have to make in “the next lessons”. Due to the sequence of IM 6–8 Math Accelerated, these choices are in the next activities and the following optional lesson.

Launch

Arrange students in groups of 2. Provide access to straightedges.

Reading, Writing: Math Language Routine 6 Three Reads. This is the first time Math Language Routine 6 is suggested as a support 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 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.
Design Principle(s): Support sense-making

How It Happens:

  1. Use this routine to support reading comprehension of this word problem without solving it for students. In the first read, students read the problem with the goal of comprehending the situation.

    Invite a student to read the problem aloud while everyone else reads with them, and then ask, “What is this situation about?” Be sure to display the two graphs or ask students to reference them while reading. Allow one 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.”

  2. 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 different student to read to the class, and then prompt students by asking: “What can be counted or measured in this situation?” Give students one minute of quiet think time, followed by another minute to share with their partner. A typical written 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 \(\frac12\) liters per minute.”

  3. In the third read, students brainstorm possible 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, and follow with the questions. Instruct students to think of ways to approach the questions without actually solving the problems.

    Consider using these questions to prompt students: “How would you approach this question?”, “What strategy or method would you try first?”, and “Can you think of a different way to solve it?”.

    Give students one minute of quiet think time followed by another minute to discuss with their partner. 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....”.

  4. As partners are discussing their solution strategies, select 1–2 students for each question 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.

    Listen for quantities that were mentioned during the second read, and take note of approaches in which the students distinguish between the graphs with differently-scaled axes.

  5. Post the summary where all students can use it as a reference.

Student Facing

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 \(\frac12\) liters per minute. The relationship between its volume of water and time can be described by the equation \(v=\frac12t\), where \(t\) is the time in minutes and \(v\) 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.
graph, horizontal axis, time in minutes, scale 0 to 80, by 20's. vertical axis, volume in liters, 0 to 50, by 10's.
  1. Sketch and label a graph of the relationship between the volume of water \(v\) and time \(t\) for Tank B on each of the axes.
  2. Answer the following questions and say which graph you used to find your answer.

    1. After 30 seconds, which tank has the most water?
    2. At approximately what times do both tanks have the same amount of water?
    3. At approximately what times do both tanks contain 1 liter of water? 20 liters?

Student Response

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Student Facing

Are you ready for more?

A giant tortoise travels at 0.17 miles per hour and an arctic hare travels at 37 miles per hour. 

  1. Draw separate graphs that show the relationship between time elapsed, in hours, and distance traveled, in miles, for both the tortoise and the hare.
     
  2. Would it be helpful to try to put both graphs on the same pair of axes? Why or why not?
     
  3. The tortoise and the hare start out together and after half an hour the hare stops to take a rest. How long does it take the tortoise to catch up?
     

Student Response

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Activity Synthesis

Begin the discussion by asking students:

  • “What question can you answer using the second graph that you can’t with the first?” (You can see when the two tanks have the same amount of water on the second graph.)
  • “Is the first graph deceptive in any way?” (Yes, it looks like Tank A will always have more volume than Tank B.)
  • “Which scale do you prefer?”

Tell students that if they had a situation where they needed to make a graph from scratch, it is important to check out what questions are asked. Some things to consider are:

  • How large are the numbers in the problem? Do you need to go out to 10 or 100?
  • What will you count by? 1s? 5s? 10s?
  • Should both axes have the same scale?

In the next lessons, students will have to make these types of choices. Tell students that while it can seem like a lot of things to keep in your head, they should always remember that there are many good options when graphing a relationship so they shouldn’t feel like they have to make the same exact graph as someone else.

2.4: Representations of Proportional Relationships (15 minutes)

Activity

The purpose of this activity is for students to graph a proportional relationship when given a blank pair of axes. They will need to label and scale the axes appropriately before adding the line representing the given relationship. In each problem, students are given two representations and asked to create two more representations so that each relationship has a description, graph, table, and equation. Then, they explain how they know these are different representations of the same situation (MP3). In the next lesson, students will use these skills to compare two proportional relationships represented in different ways.

Identify students making particularly clear graphs and using situation-appropriate scales for their axes. For example, since the second problem is about a car wash, the scale for the axis showing the number of cars does not need to extend into the thousands.

Launch

Arrange students in groups of 2. Provide access to straightedges. Give 3 minutes of quiet work time for students to begin the first problem and then tell students to check in with their partners to compare tables and how they are labeling and scaling the axes. Ask students to pause their work and select a few students to share what scale they are using for the axes and why they chose it. It is important to note that the scale chosen should be reasonable based on the context. For example, using a very small scale for steps taken does not make sense.

Give partners time to finish the remaining problems and follow with a whole-class discussion.

Action and Expression: Internalize Executive Functions. Provide students with grid or graph paper to organize their work with creating a table and graph for the situation.
Supports accessibility for: Language; Organization

Student Facing

  1. Here are two ways to represent a situation.

    Description:

    Jada and Noah counted the number of steps they took to walk a set distance. To walk the same distance, Jada took 8 steps while Noah took 10 steps. Then they found that when Noah took 15 steps, Jada took 12 steps.

    Equation:

    Let \(x\) represent the number of steps Jada takes and let \(y\) represent the number of steps Noah takes. \(\displaystyle y=\frac54x\)

    1. Create a table that represents this situation with at least 3 pairs of values.

    2. Graph this relationship and label the axes.

      Blank coordinate plane.
    3. How can you see or calculate the constant of proportionality in each representation? What does it mean?

    4. Explain how you can tell that the equation, description, graph, and table all represent the same situation.

  2. Here are two ways to represent a situation.

    Description:

    The Origami Club is doing a car wash fundraiser to raise money for a trip. They charge the same price for every car. After 11 cars, they raised a total of $93.50. After 23 cars, they raised a total of $195.50.

    Table:

    number of
    cars
    amount raised
    in dollars
    11 93.50
    23 195.50
    1. Write an equation that represents this situation. (Use \(c\) to represent number of cars and use \(m\) to represent amount raised in dollars.)
    2. Create a graph that represents this situation.
      quadrant 1 grid. horizontal axis, c. 25 units. vertical axis, m. 10 units. 
    3. How can you see or calculate the constant of proportionality in each representation? What does it mean?
    4. Explain how you can tell that the equation, description, graph, and table all represent the same situation.

Student Response

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Activity Synthesis

Ask previously identified students to share their graphs and how they chose the scales for their axes. If possible, display several graphs from each question for all to see as students share.

Ask students “Which representation makes it more difficult (and less difficult) to calculate the constant of proportionality? Why?” and give 1 minute of quiet think time. Invite several students to share their responses.

Tell students that the constant of proportionality can be thought of as the rate of change of one variable with respect to the other. In the case of Jada and Noah, the rate of change of \(y\), the number of steps Noah takes, with respect to \(x\), the number of steps Jada takes, is \(\frac54\) Noah steps per Jada steps. In the case of the Origami Club’s car wash, the rate of change of \(m\), the amount they raise in dollars, with respect to \(c\), the number of cars they clean, is 8.50 dollars per car.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine for students to respond in writing to the final question, “Explain how you can tell that the equation, description, graph, and table all represent the same situation.” Give students time to meet with 2–3 partners, to share their responses and get feedback. Encourage the listener to ask clarifying questions such as, “How did you identify the same constant of proportionality?” or “Did the scales for the axes cause any confusion?” Have the students write a final draft based on their peer feedback. This will help students to generalize the process for identifying the same constant of proportionality.
Design Principle(s): Optimize output; Cultivate conversation

2.5: Info Gap: Proportional Relationships (15 minutes)

Activity

This info gap activity gives students an opportunity to determine and request the information needed when working with proportional relationships. In order to graph the relationship and the requested information, students need to think carefully about how they can scale the axes.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then asking for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of one of the cards for reference and planning:

Problem card with grid and data card with data

Listen for how students request (and supply) information about the relationship between the two ingredients. Identify students using different scales for their graphs that show clearly the requested information to share during the discussion.

Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Consider using this activity instead of Representations of Proportional Relationships if students have demonstrated a strong understanding of graphing and proportional relationships.

Launch

Tell students that they will continue their work graphing proportional relationships. Explain the Info Gap and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. Provide access to straightedges. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for the second problem and instruct them to switch roles.

Representation: Provide Access for Perception. Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity.
Supports accessibility for: Language; Memory
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to choose a scale and graph a proportional relationship. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?”
Design Principle(s): Cultivate Conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to be able to answer the question.

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

    If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

  3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

  4. Read the problem card and solve the problem independently.

  5. Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Student Response

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Student Facing

Are you ready for more?

Ten people can dig five holes in three hours. If \(n\) people digging at the same rate dig \(m\) holes in \(d\) hours:

  1. Is \(n\) proportional to \(m\) when \(d=3\)?
  2. Is \(n\) proportional to \(d\) when \(m=5\)?
  3. Is \(m\) proportional to \(d\) when \(n=10\)

Student Response

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Anticipated Misconceptions

Some students may be unsure how large to make their scale before they answer the question on the card. Encourage these students to answer the question on their card and then think about how to scale their graph.

Activity Synthesis

After students have completed their work, ask previously identified students to share their graphs and explain how they chose their axis. Some guiding questions:

  • “Other than the answer, what information would have been nice to have?”
  • “How did you decide what to label the two axes?”
  • “How did you decide to scale the horizontal axis? The vertical?”
  • “What was the rate of change of grams of honey per cups of flour? Where can you see this on the graph you made?” (4.5 grams of honey per cup of flour.)
  • “What was the rate of change of grams of salt per cups of flour? Where can you see this on the graph you made?” (2.5 grams of salt per cups of flour)

Lesson Synthesis

Lesson Synthesis

Display this blank graph for all to see and provide pairs of students with graph paper.

quadrant 1

Ask pairs to draw a copy of the axis and give a signal when they have finished. (You may need to warn students 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 \(y=6x\). After students graph, add the line representing the equation to the graph on display. Then, ask students to make a second graph with the same horizontal scale, but with a vertical scale that makes \(y=6x\) 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 \(y=6x\) 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 we might want to graph this relationship with such a large vertical scale?” (If we needed to also graph something like \(y=60x\), we would need a pretty big vertical scale in order to see both lines.)

If time allows, here are some questions for discussion:

  • “The proportional relationship \(y=5.5x\) includes the point \((18,99)\) on its graph. How could you choose a scale for a pair of axes with a 10 by 10 grid to show this point?” (Have each grid line represent 10 or 20 units.)
  • “What are some things you learned about graphing today that you are going to try to remember for later?”

2.6: Cool-down - Graph the Relationship (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

The scales we choose when graphing a relationship often depend on what information we want to know. For example, say two water tanks are filled at different constant rates. The relationship between time in minutes \(t\) and volume in liters \(v\) of tank A is given by \(v=2.2t\).

For tank B the relationship is \(v=2.75t\)

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. 
graph, horizontal axis, time in minutes, scale 0 to 90, by 10's. vertical axis, volume in liters, 0 to 140, by 10's. 

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.

What representation we choose for a proportional relationship also depends on our purpose. When we create representations we can choose helpful values by paying attention to the context. For example, if Tank C fills at a constant rate of 2.5 liters per minute, we could write the equation \(v=2.5t\). If we want to compare how long it takes Tanks A, B, and C to reach 110 liters, then we could graph them on the same axis. If we want to see the change in volume every 30 minutes, we could use a table:

minutes (t) liters (v)
0 0
30 75
60 150
90 225

No matter the representation or the scale used, the constant of proportionality, 2.5, is evident in each. In the equation it is the number we multiply \(t\) by. In the graph it is the slope, and in the table it is the number by which we multiply values in the left column to get numbers in the right column. We can think of the constant of proportionality as a rate of change of \(v\) with respect to \(t\). In this case the rate of change is 2.5 liters per minute.