Lesson 2

In the previous warm-up, students compared different proportional relationships on two sets of axes that were scaled the same. In this warm-up, students will work with two sets of axes scaled differently and the same proportional relationship. The purpose of this warm-up is to make explicit that the same proportional relationship can appear to have different steepness depending on the axes, which is why paying attention to scale is important when making sense of graphs or making graphs from scratch. Students learned how to graph and write equations for proportional relationships in previous grades, so this warm-up also helps refresh those skills.

Identify students using different strategies to graph the relationship on the new axes. For example, since this is a proportional relationship, some students may scale up the point \((8,14)\) to something like \((40,70)\) and then plot that point before drawing a line through it and the point \((0,0)\). Other students may use the equation they wrote and one of the \(x\)-values marked on the new axes to find a point on the line and draw in the line from there.

Give 2–3 minutes of quiet work time followed by a whole-class discussion.

Display the two images from the activity for all to see. Invite previously identified students to share how they graphed the relationship on the new line.

Ask students, “Which graph looks steeper to you?” Students may see the first line as steeper, even though the two lines have the same slope. It is important students understand that this is one of the reasons mathematics uses numbers (slope) to talk about the steepness of lines and not “looks like.”

The purpose of this activity is for students to identify the same proportional relationship graphed using different scales. Students will first sort the cards based on what proportional relationship they represent and then write an equation representing each relationship. Identify and select groups using different strategies to match graphs to share during the Activity Synthesis. For example, some groups may identify the unit rate for each graph in order to match while others may choose to write equations first and use those to match their graphs.

Arrange students in groups of 4. Provide each group with a set of 12 pre-cut slips.

If students have trouble recalling the meaning of the constant of proportionality, remind them that one way to think about it is “the change in \(y\) for every unit change in \(x\).” 

As a result of this conversation, students should understand that the scale of the axes a graph is drawn on can hide the actual relationship between the two variables if you just look at the steepness of the line without paying attention to the numbers on the axes.

Ask previously selected groups to share their strategies for matching the graphs. Highlight uses of relevant vocabulary such as “constant of proportionality” or “unit rate.”

Have students look at Card A. Ask students, “Do you think this graph looks like \(y=\frac14x\)? Why or why not?” Possible reasons from students are:

• No, I think this graph looks like \(y=x\).
• I would expect \(y=\frac14x\) to be less steep since an increase in \(x\) means \(\frac14\) as much of an increase in \(y\).
• Since the \(x\)-scale is four times the \(y\)-scale, the line looks steeper that I expected.

Give students a moment to identify another card that they think does not “look like” the equation and select a few students to share the graph they chose and explain their thinking.

Building off the work in the previous activity, students now graph a proportional relationship on two differently scaled axes and compare the proportional relationship to an already-graphed non-proportional relationship on the same axes. Students are asked to make sense of the intersections of the two graphs by reasoning about the situation and consider which scale is most helpful: the zoomed in, or the zoomed out. In this case, which graph is most helpful depends on the questions asked about the situation.

Arrange students in groups of 2. Provide access to straightedges.

Begin the discussion by asking students:

• “What question can you answer using the second graph that you can’t with the first?” (You can see when the two tanks have the same amount of water on the second graph.)
• “Is the first graph deceptive in any way?” (Yes, it looks like Tank A will always have more volume than Tank B.)
• “Which scale do you prefer?”

Tell students that if they had a situation where they needed to make a graph from scratch, it is important to check out what questions are asked. Some things to consider are:

• How large are the numbers in the problem? Do you need to go out to 10 or 100?
• What will you count by? 1s? 5s? 10s?
• Should both axes have the same scale?

In the next lessons, students will have to make these types of choices. Tell students that while it can seem like a lot of things to keep in your head, they should always remember that there are many good options when graphing a relationship so they shouldn’t feel like they have to make the same exact graph as someone else.