This activity provides the first example of a relationship that is not proportional. The second question focuses students’ attention on the unit rates. If the relationship were proportional then regardless of the number of people in a vehicle, the cost per person would be the same. The question about the bus is to show students that they can’t just scale up from 10. Students who write an equation also see that it is not of the form . In a later lesson students will learn that only equations of this form represent proportional relationships.

Monitor for students who approached this problem using different representations.

The goal of this discussion is to highlight how we know that the relationship is not proportional. Select students to explain why they think the relationship is or is not proportional. Some reasons they could give include:

• The cost per person is different for different numbers of people in a vehicle. In other words, the quotients of the values in each row are not equal for all rows of the table.
• The ratios of people in the vehicle to total entrance cost are not equivalent. We can't multiply the entries in one row by a scale factor to get the entries in another row.
• We can't multiply the entries in the first column by the same number (constant of proportionality) to get the numbers in the second column.

Students who found an equation will also note that the equation is not of the same form as other equations, but they can’t use this as a criterion until the class has established that only equations of this form represent proportional relationships. (This part of the discussion should come at the end of the next lesson, after students have analyzed lots of different equations.)

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question by correcting errors, clarifying meaning, and adding details.

• Display this first draft:

  “Yes, the relationship is proportional. It says it costs $2 per person, so the constant of proportionality is 2.”

  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 purpose of this activity is to understand that discrete values in a table can be used to know for sure that a relationship is not proportional. However, they can’t be used to know for sure that a relationship definitely is proportional. There could be other values in the relationship, which are not shown on the table, that don’t fit the pattern.

This activity builds on previous ones involving constant speed but it analyzes pace (minutes per lap) rather than speed (laps per minute). Explaining why the information given in the table is enough to conclude that Han didn't run at a constant pace but is not enough to know for sure whether Clare ran at a constant pace requires students to make a viable argument (MP3).

Students are likely to answer that Clare is running at a constant pace because the minutes per lap shown in the table are the same for each lap. Because we only have four data points in a table, spaced at 5-minute intervals, Clare could still be speeding up and slowing down between the recorded times. However, given the data, it is reasonable to assume Clare is running at a constant pace for the purpose of estimating times or distances.

Invite students to explain why they think each person is or is not running at a constant pace. Point out to students that although the data points in the table for Clare are pairs in a proportional relationship, these four pairs of values do not guarantee that Clare ran at a constant pace. She might have, but it is unknown if she was running at a constant pace between the times that the coach recorded.

Ask the following questions:

• “Can you represent either relationship with an equation?” (The answer for Han is “no” and the answer for Clare is “yes, if she really ran at a constant pace between the points in time when the times were recorded.”)
• “What equation can we write to represent Clare’s pace in minutes per lap?” ()
• “In the table for Clare's run, do the pairs of values still form a proportional relationship if we calculate laps per minute instead of minutes per lap?” (yes)
• “What equation can we write to represent that relationship in laps per minute?” ()