This activity builds on “Tyler’s Walk“ from the previous lesson, as students compare the time-distance relationships for three more people who travel from the ticket booth to the bumper cars. Students create tables and graphs that represent each relationship. They relate the constant of proportionality for each relationship to the person’s speed in meters per second.

Students also identify that one of the relationships is not proportional. Because Mai did not start walking at the same time as the others, the line representing her walk does not go through .

In the digital version of the activity, students use an applet to graph multiple relationships on the same coordinate plane. 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. If students don't have individual access, displaying the applet for all to see would be helpful during the synthesis.

The goal of this discussion is to highlight how the constant of proportionality for each relationship is illustrated on the graph and to connect this to each person’s speed in meters per second. First, invite students to share their answer to the last question, about who was moving the most quickly. (Lin, because she got to the bumper cars in the least amount of time. She traveled the most meters per second.)

Next, help students make connections between how the constant of proportionality is shown in the different representations. Consider asking:

• "For each graph that shows a proportional relationship, what is the constant of proportionality?" (Tyler: 1.25, Diego:, Lin: 2.5)
• "Where can we see the constants of proportionality in the tables?” (the values in the second column that correspond to the value of 1 in the first column)
• “Where can we see the constants of proportionality on the graphs?" (the -coordinate of points where the -coordinate is 1)
• "Which graph does not represent a proportional relationship?" (Mai's graph does not pass through the origin, so it does not represent a proportional relationship. That is, the distance she traveled is not proportional to the time elapsed since Tyler left the ticket booth.)
• “How could we represent the three proportional relationships with equations?” (Tyler: , Diego: , Lin: , or equivalent)
• “An equation that represents Mai’s journey is . What do you notice about this equation compared to the other three?” (This equation is not of the form , while the other three equations are.)

In this activity students continue to interpret graphs that show more than one proportional relationship on the same set of axes. To increase the need for abstract reasoning, the graphs in this activity do not include numerical labels. Students can reason abstractly (MP2) by picking an arbitrary time and comparing the corresponding distances or by picking an arbitrary distance and comparing the corresponding times. The goal is for students to realize that when two proportional relationships are shown on the same set of axes, the steeper graph has the greater constant of proportionality, regardless of how the axes are scaled. As students notice these connections, they are making use of structure (MP7).

As students work on the first problem, monitor for groups who use these different approaches:

• Compare the distances each object travels in the same amount of time, such as in the graph with the vertical dashed

• Compare the times it takes each object to travel the same distance, such as in the graph with the horizontal dashed line
• Recognize that the steeper line has the greater constant of proportionality and interpreting what this means in context of the situation

Plan to have students present in this order to support moving them from more concrete to more abstract reasoning.

In the digital version of the activity, students use an applet to interpret and compare two proportional relationships. The applet allows students to watch the graphs animated over time, as well as add a vertical or horizontal line to facilitate comparing the graphs at a specific time or distance, respectively. 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, displaying the applet for all to see would be helpful during the synthesis.

The purpose of this discussion is to help students interpret what it means that one graph is steeper than another for cases where there are two different proportional relationships graphed on the same pair of axes. It is important that students do not assume “steeper always means faster,” but that they understand why it is in this case by reasoning abstractly and attending to the referents for points on the graphs. If the same relationships were graphed with distance on the horizontal axis and time on the vertical axis, a steeper line would indicate a slower speed. If the same relationships were graphed on separate axes, their scales could be different. Because the graphs share the same axes, it is implicit that comparisons between them occur relative to the same units.

Invite previously selected students to share their reasoning for the first problem. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How does each method tell us which object is traveling at a greater speed?”
• “How does each method deal with the fact that there are no numbers labeled on the graphs?”

The key takeaways are:

• When comparing proportional relationships, it’s important to interpret the constants of proportionality in context of the situation.
• A steeper graph does not always mean that the object is moving faster.
  • If the relationships are on two separate graphs, then the scales on the axes could be different.
  • If distance is on the horizontal axis and time is on the vertical axis, a steeper line indicates a slower speed.
  • If the quantities are something other than distance and time, then the comparison is of some other aspect besides speed.