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.
In the digital version of the activity, students use an applet to plot points 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.
Student Lesson in Spanish
Launch
Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Activity
None
Plot the points .
What do you notice about the graph?
Student Response
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Building on Student Thinking
Some students may reverse the - and -coordinates when plotting points, such as plotting and instead of and . Direct their attention to the and labels in the image and clarify that an ordered pair is written .
Activity Synthesis
The goal of this discussion is to review how to graph ordered pairs, , on the coordinate plane. Invite students to share their observations about the graph. 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.
MLR8 Discussion Supports. Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “The points are straight” as ”The points line up on the coordinate plane so that they could all be connected with a single line.” Advances: Representing
10.2
Activity
Standards Alignment
Building On
6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
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.
Launch
Arrange students in groups of 2. Provide access to rulers. Give students 5 minutes of quiet work time followed by partner and whole-class discussion.
If students are unsure how to plot points from the table, consider rewriting the values in the first row of the table as an ordered pair, , and demonstrating how to plot this point.
Activity
None
Activity Synthesis
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.
MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner. Advances: Listening, Speaking
10.3
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2
Recognize and represent proportional relationships between quantities.
In this activity, students sort different graphs and tables that represent situations they have worked with during previous activities. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Monitor for different ways groups choose to categorize the representations, but especially for categories that distinguish between proportional and nonproportional relationships. The purpose of this activity is to illustrate the idea that the graph of a proportional relationship is a line through the origin, though students will not have the tools for a formal explanation until grade 8.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.
Arrange students in groups of 2 and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
Give students 2 minutes to sort the cards into categories of their choosing.
Pause for a class discussion, even if some groups haven’t finished their sorting.
Select groups to share their categories and how they started sorting their cards. If possible, select groups to share their categories in this order:
Separating the tables from the graphs
Matching pairs of a table and a graph that represent the same relationship
Separating the proportional relationships from the nonproportional relationships
Discuss as many different types of categories as time allows.
Attend to the language that students use to describe their categories, giving them opportunities to describe the cards more precisely. Highlight the use of terms like “table,” “graph,” “proportional,” “nonproportional,” “straight line,” and “corresponds.”
After a brief discussion, invite students to complete the remaining questions.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory
Activity
None
Your teacher will give you a set of cards that show representations of relationships.
Sort the cards into categories of your choosing. Be prepared to describe your categories.
Pause for a whole-class discussion.
Take turns with your partner to match a table with a graph.
For each match that you find, explain to your partner how you know it’s a match.
For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Which of the relationships are proportional?
What do you notice about the graphs of proportional relationships? Do you think this will hold true for all graphs of proportional relationships?
Activity Synthesis
The goal of this discussion is to generalize that for all proportional relationships, the points lie on a straight line through the origin. Direct students' attention to the reference created using Collect and Display. Ask students to share how they determined which relationships were proportional. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.
Ensure that students are in agreement that Cards B, C, D, F, and I are the graphs of proportional relationships. Then, invite 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 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? (It makes sense to draw an unbroken line for 7F and 8I. The rest should use dots that are not connected.) 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.
Math Community
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
“I noticed our norm “” in action today, and it really helped me/my group because .”
Lesson Synthesis
Share with students “Today we examined graphs of relationships. Some were proportional and some were not.”
To help students generalize about graphs of proportional relationships, consider asking students:
“What characteristics were shared by all the graphs of proportional relationships that we saw?” (The points were arranged in a straight line. The point lines up with the other points.)
“What characteristics might you see on a graph that would let you know that the relationship is not proportional?” (The points are not arranged in a straight line. The point does not line up with the other points.)
If desired, use the following quick activity to review these concepts. Tell students, “Sketch a quick graph of a relationship that is or isn’t proportional and trade sketches with your partner. Then, explain to your partner why their graph does or does not show a proportional relationship.”
Student Lesson Summary
One way to represent a proportional relationship is with a graph. Here is a graph that represents different amounts that fit the situation, “Blueberries cost \$6 per pound.”
Line graph. Weight in pounds. Cost in dollars. Horizontal axis, 0 to 5, by 1's. Vertical Axis, 0 to 40, by 10's. Line begins at origin, trends upward and right, passes through 1 comma 6, 2 comma 12, 3 comma 18, 4 point 5 comma 27.
Different points on the graph tell us, for example, that 2 pounds of blueberries cost \$12, and 4.5 pounds of blueberries cost \$27.
Sometimes it makes sense to connect the points with a line, and sometimes it doesn’t. For example, we could buy 4.5 pounds of blueberries or 1.875 pounds, or any other part of a whole pound. So all the points between the whole numbers make sense in the situation, and any point on the line is meaningful.
If the graph represented the cost for different numbers of sandwiches (instead of pounds of blueberries), it might not make sense to connect the points with a line, because it is often not possible to buy 4.5 sandwiches or 1.875 sandwiches. Even if only points make sense in the situation, though, sometimes we connect them with a line anyway to make the relationship easier to see.
Graphs that represent proportional relationships all have a few things in common:
Points that satisfy the relationship lie on a straight line.
The line that they lie on passes through the origin, .
Here are some graphs that do not represent proportional relationships:
Graph of a non-proportional relationship, x y plane, origin O. Horizontal axis scale 0 to 7 by 1’s. Vertical axis scale 0 to 6 by 1’s. There are points at: (0 comma 0), (1 comma 1), (2 comma 3), (3 comma 4), (4 comma 4 point 5), (5 comma 5), (6 comma 5 point 1), and (7 comma 5 point 2).
These points do not lie on a line.
Line graph. Horizontal axis, 0 to 7, by 1's. Vertical Axis, 0 to 6, by 1's. Line begins on y axis at 0 comma 2, trends upward and right, passes through 2 comma 3, 4 comma 4, 6 comma 5.
This is a line, but it doesn’t go through the origin.
Standards Alignment
Building On
5.G.A
Graph points on the coordinate plane to solve real-world and mathematical problems.
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Is there a proportional relationship between and ?
Plot the pairs in the table on the coordinate plane.
What do you notice about the graph?
Student Response
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Building on Student Thinking
Student Response
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Building on Student Thinking
If students struggle to get started 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.
