Lesson 10

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.

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.

Students work in pairs to match tables to graphs and to practice articulating their reasoning (MP3). This task is intended to foster understanding of correspondences between tables and graphs.

Students sort the graphs and justify their sorting schemes. Then, they compare the way they sorted their graphs with a different group. The purpose of this activity is to illustrate the idea that the graph of a proportional relationships is a line through the origin. Students will not have the tools for a formal explanation until grade 8.

Demonstrate how the matching activity works and how to have mathematical dialogue about the decisions being made (see the instructions in the task statement). When students finish the activity, they use an answer key to check their answers. If adjustments need to be made, students discuss any errors they made.

While students are working to match graphs to written descriptions and tables, circulate and ask them to justify their choices.

Arrange students in groups of 2. Place copies of answer keys in envelopes.

Ask students to observe similarities and differences in the graphs and to create a rationale for sorting them. Provide access to sticky notes. If necessary, specify how the groups will trade places after they finish sorting their graphs into categories.

Demonstrate how to set up and conduct the matching activity. Choose a student to act as your partner. Mix up the cards and place them face-up. Point out that the cards contain either tables or graphs. Select one of each style of card and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree, e.g. by explaining your mathematical thinking, asking clarifying questions, etc.

Give each group cut-up slips for matching. Tell students to check their matches after they complete the activity using the answer keys.

If students struggle to get starting making any matches, ask questions like “How would we expect this row in the table to look on the graph?” Or, “See this point on the graph? What corresponds to it in the table?”

A common misunderstanding is to assume that if the points lie on a line, then the graph represents a proportional relationship. Ask questions about the table to assist students in realizing the error.

Ask students how they determined which relationships were proportional. Invite volunteers to share the proportional relationships to ensure common understanding. Address any discrepancies through questioning: “How do you know the relationship is proportional? What have you learned about proportional relationships that applies here?”

Select students to share what they noticed about the characteristics of graphs of proportional relationships. Some observations might conclude:

• Points whose coordinates satisfy the relationship lie on a line.
• The line passes through the point \((0,0)\).

This would be a good place to either introduce the term origin to refer to the point \((0,0)\) (or to remind students of it, if they have encountered it before).

If time permits, discuss which written descriptions of proportional relationships would warrant “connecting the dots.” In other words, which proportional relationships are best represented with dots, and which are best represented with an unbroken line? Of the cards which describe a proportional relationship, it makes sense to draw an unbroken line for 7 and 8. The rest should use dots that are not connected. None include negative values without some assumptions (e.g., scenarios that involve owing money for card 2). Students should realize that even when the graph of a proportional relationship is represented by unconnected points, they lie on a line through the origin.