In this activity students interpret points on the graph of a proportional relationship in terms of what they mean about the situation (MP2). This activity is intended to further students’ understanding of the graphs of proportional relationships in the following respects:

• For any point on the graph except , the quotient of the coordinates, , is the constant of proportionality.
• When the -coordinate is 1, the corresponding -coordinate is , the constant of proportionality.

Students explain correspondences between parts of the table and parts of the graph. The graph is simple so that students can focus on what a point means in the situation represented. Students need to realize, however, that the axes are marked in 10-unit intervals.

The goal of the discussion is to make connections between the table and the graph and how they each represent the situation. First, ask students:

• "How far is the ticket booth from the bumper cars? How do you know?" (50 meters, because the point represents his arrival at the bumper cars. 40 is the elapsed time in seconds and 50 is the distance in meters.)

Consider clarifying for students that this is assuming that Tyler walked in a straight line. This is an opportunity for attention to precision (MP6) and making explicit assumptions about a situation (MP4).

For each of the following questions, ask students to share how they can tell the answer from the table and how they can tell from the graph.

• “What quantities are shown?” (distance in meters that Tyler is from the ticket booth and elapsed time in seconds since he started walking)
  • “How can you tell from the table?” (by looking at the column headers)
  • “How can you tell from the graph?” (by looking at the axes labels)
• “What does each pair of values mean? For example and ?” (Tyler was 25 meters from the ticket booth after 20 seconds. Tyler was 0 meters away from the ticket booth after 0 seconds.)
  • “How can you tell from the table?” (The value in the first column gives the elapsed time in seconds since Tyler started walking. The value in the second column tells how many meters away from the ticket booth Tyler was at that time.)
  • “How can you tell from the graph?” (The -coordinate gives the elapsed time in seconds since Tyler started walking. The corresponding -coordinate shows how many meters away from the ticket booth Tyler was at that time.)
• “Is the relationship proportional?” (yes)
  • “How can you tell from the table?” (Dividing the values on any row, except , gives the same unit rate.)
  • “How can you tell from the graph?” (The points lie on a straight line through .)
• “What is the constant of proportionality?” (1.25)
  • “How can you tell from the table?” (Find the value for the second column that goes next to the 1 in the first column.)
  • “How can you tell from the graph?” (Find the point that lies on the same line as the other points or find the quotient for any point on the graph besides .)

After students have seen how the different representations show the same information, consider asking students, “Are there any benefits or drawbacks to one representation compared to the other? Which representation do you prefer?”

Lastly, ask students to write an equation for this proportional relationship. (Sample responses: or )

In this activity students create the graph of a proportional relationship given only one pair of values. Then they use the graph to find the constant of proportionality and interpret it in context. 

In previous activities students were given a table with multiple pairs of values to graph. Being given only one pair of values here reinforces the idea that the graph of any proportional relationship makes a straight line through the origin. This activity asks students to connect the points with a solid line. The class discussion revisits the idea that people often connect discrete points with a line to make the relationship more clear, even when the in-between values don’t make sense.

In the digital version of the activity, students use an applet to graph a proportional relationship on the coordinate plane. The applet allows students to plot points and draw lines. The digital version may help students graph quickly and accurately so they can focus more on the mathematical analysis.

Invite students to share their value and interpretation of . Ask them for different ways to express this information. (Each seagull eats 2.5 pounds of garbage. Or the rate of garbage consumption is 2.5 pounds per seagull.)

Ask students:

• “Is it possible to interpret the meaning of each point on the solid line?” (No, only whole numbers of seagulls make sense.)
• “Why is it still useful to draw in the line? How can it help us to learn more about the situation?" (It helps us to easily find out how much garbage different numbers of seagulls eat. It also helps us to estimate the value of .)