Lesson 2

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.

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

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.”

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.

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

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 \(x\).” 

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.

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.

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.

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

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.

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.

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.

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.

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:

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.

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.

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.

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)