The purpose of this activity is for students to identify the same proportional relationship graphed using different scales. Students will first sort the graphs based on what proportional relationship they represent and then write an equation representing each relationship. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7).

Monitor for and select groups using different strategies to match graphs to share later. 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.

The goal of this discussion is for students to understand that looking only at the steepness of the line without paying attention to the numbers on the axes can hide the actual relationship between the two variables.

Once all groups have completed the Card Sort, ask previously selected groups to share their strategies for grouping the graphs. Discuss the following:

• “Which graphs were tricky to group? Explain why.”
• “Did you need to make adjustments in your groups? What might have caused an error? What adjustments were made?”

In this activity, students graph a proportional relationship on two differently scaled axes and compare the proportional relationship to an already-graphed nonproportional relationship on the same axes. Students make sense of the intersections of the two graphs by reasoning about the situation and consider which scale is most helpful: zoomed in, or zoomed out. In this case, which graph is most helpful depends on the questions asked about the situation (MP4).

This is the first time Three Reads (Math Language Routine 6) is suggested 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 important 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.

The goal of this discussion is to emphasize how selecting an appropriate scale is important when creating a graph from scratch. Begin the discussion by asking students:

• “What information can be seen using the second graph that can’t be seen with the first?” (Tank A starts out with more water than Tank B at first, and then with less water than Tank B later on. The two tanks have the same amount of water after 60 minutes.)
• “If someone only looked at the first graph, what might they think (incorrectly)?” (The first graph makes it look like Tank A will always have more water than Tank B.)
• “In what situation might the first graph be more useful?” (If someone wanted to answer a question about the tanks after only 1 or 2 minutes.)

Explain that if making a graph from scratch, it is important to first check what questions are being asked. Some things to consider are:

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

This activity gives students a chance to choose an appropriate scale when graphing a proportional relationship on a given set of blank axes (MP4). Monitor for students who create particularly clear graphs using situation appropriate scales. For example, since the problem is about a car wash, the scale for the axis showing the number of cars does not need to extend into the thousands.

The purpose of this discussion is to introduce students to the term “rate of change.” Begin by inviting 2–3 students to share the graphs they created. Emphasize how different scales can be used, but in order to be helpful, the scale for the number of cars, , on the horizontal axis should extend to at least 23 and the scale for the amount raised in dollars, , on the vertical axis should extend to at least 200.

Next, tell students that an equation that represents this situation is , where is the number of cars, and is the total dollars raised. Display this equation for all to see, then discuss:

• “What is the constant of proportionality and what does it mean?” (The constant of proportionality is 8.5 and it means that each car washed raised $8.50.)

• “How can you see the constant of proportionality in the graph and the table? (Graph: The slope of the line is equivalent to 8.5. Table: For any given row, the amount raised in dollars divided by the number of cars washed equals 8.5.)

• “Which representation do you think is more useful when calculating the constant of proportionality? Why?”

Explain that the constant of proportionality can be thought of as the rate of change: the amount one variable changes by when the other variable increases by 1. In the case of the Origami Club’s car wash, the rate of change of , the amount they raise in dollars, with respect to , the number of cars they wash, is 8.50 dollars per car.

In this activity, students graph a proportional relationship but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask 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).

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “How did you decide what to label the two axes?”

• “How did you decide to scale the horizontal axis? The vertical axis?”

• “Where can you see the rate of change of grams of honey per cups of flour on the graph?” 

• “Where can you see the rate of change of grams of salt per cups of flour on the graph?”