In this activity, students create a graph to represent the proportional relationship given in a table. The goal is for students to notice that the points lie on a straight line that goes through the origin. The class discussion also prompts students to consider whether it makes sense to connect the points with a line.

In the digital version of the activity, students use an applet to plot points from a table on the coordinate plane. The applet allows students to add, remove, adjust, and label points. The digital version may help students graph quickly and accurately so they can focus more on the mathematical analysis.

The goal of this discussion is to highlight the fact that the graph of a proportional relationship makes a straight line through the origin. Display a graph with the points plotted correctly for all to see. Invite students to share how to label the axes. (The -axis represents “number of T-shirts” and the -axis represents “cost in dollars”.)

Ask students to share their observations about the plotted points. If not mentioned by students, highlight that the points lie on a straight line and that the line goes through .

Direct students’ attention to considering the meaning of other points that are also on this line but were not in the table. Ask, “Could we buy 0 shirts? 7 T-shirts? 10 T-shirts? Can we buy half of a T-shirt?” Note that the graph consists of discrete points because only whole numbers of T-shirts make sense in this context. However, people often connect discrete points with a line to make the relationship more clear, even when the in-between values don’t make sense.

Ask the students, “Suppose instead of price per shirt, this graph displayed the cost of cherries that are $8 per pound. Given that context, how should we change the graph?” Weights need not have integer values, so the graph is not restricted to discrete points. If you haven’t done so already, draw the ray starting at (0, 0) that passes through the points.

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 )