Lesson 7

This warm-up prepares students for graphing proportional relationships in the coordinate plane. They practice graphing coordinate points and notice that all points lie on a straight line. 

Invite students to share their observations. Ask if other students agree. If some students do not agree that the points lie on a straight line, ask which points break the pattern and give students a chance to self-correct their work.

This introductory activity asks students to plot points using tables of values that represent scenarios familiar from previous lessons. This activity is intended as a review of the coordinate plane, its axes, and plotting ordered pairs.

Adjust this activity to 15 minutes.

During the synthesis of this activity, display these graphs of proportional and nonproportional relationships for all to see.

Graphs of Proportional Relationships

Graphs of Nonproportional Relationships

Ask students, "What properties do the graphs representing proportional relationships have?" (The points are all in a line. The line goes through the point \((0,0)\).)

If necessary, use a straightedge to show that the points are all in a line.

Introduce the word origin to refer to the point \((0,0)\) if students are unfamiliar with the term.

Discuss the first question if students had trouble with it.

Ask students to share their observations about the plotted points. 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.

This activity is intended to further students’ understanding of the graphs of proportional relationships in the following respects:

• points on the graph of a proportional relationship can be interpreted in the context represented (MP2)
• for these points, the quotient of the coordinates is—excepting \((0,0)\)—the constant of proportionality
• if the first coordinate is 1, then the corresponding coordinate is \(k\), 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 discussion questions are opportunities for students to construct viable arguments and critique the reasoning of others (MP3).

Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.

These questions can be used for discussion or for students who need scaffolding.

• "What quantities are shown in the graph?" (Distance in meters that Tyler is from the ticket booth and time elapsed in seconds since he started walking.)
• "How far is the ticket booth from the bumper cars?" (50 meters, 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).
• "Do the values in your table show a proportional relationship? How do you know?" (Based on prior lessons in this unit, students should identify the relationship as proportional because for every point the unit rate is the same.)
• "What do the coordinates of the points on the graph show?" (The first coordinate gives amount of time in seconds that elapsed since Tyler started walking. Its corresponding second coordinate shows how many meters away from the ticket booth Tyler was at the corresponding time, assuming that Tyler walked in a straight line.)

After students work on the task, it is important to discuss how the axis labels and the description in the task statement help us interpret points on the graph.

Consider asking these questions:

• "What quantities are shown in the graph?" (Distance in meters that Tyler is from the ticket booth and time elapsed in seconds since he started walking.)
• "How far is the ticket booth from the bumper cars?" (50 meters, 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).
• "Do the values in your table show a proportional relationship? How do you know?" (Based on prior lessons in this unit, students should identify the relationship as proportional because for every point the unit rate is the same.)
• "What do the coordinates of the points on the graph show?" (The first coordinate gives amount of time in seconds that elapsed since Tyler started walking. Its corresponding second coordinate shows how many meters away from the ticket booth Tyler was at the corresponding time, assuming that Tyler walked in a straight line.)

Ask students for the equation of this proportional relationship. Finally ask where students see \(k\), the constant of proportionality, in each representation, the equation, the graph, the table, and the verbal description of the situation.