If students calculate an incorrect value for the slope of a line perpendicular to , ask them to check their work by multiplying the original slope and their proposed perpendicular slope. If the result is not -1, something is wrong.

This activity gives students an opportunity to determine and request the information needed to graph a line and write the equation that represents it.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for the information they need in order to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

There are several possible equivalent equations for each Problem Card. The equations students write will depend on the information they elicit from their partner. Monitor for different solutions to highlight during the Activity Synthesis.

If students struggle to elicit enough information, suggest they ask about the relationships of their line to the existing lines on the graph. Ask them what information they need in order to write the equation of a line. Point out that there are several different ways to write an equation.

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “How did you determine the slope of the line?”
• “Did anyone use a different piece of information to come to this same conclusion?”

Invite previously selected students to share different equations for each card. Challenge students to show that the forms are all equivalent for a given line. For example, for Problem Card 2, if students used the point , their equation will look like . If students used the -intercept, , their equation will look like . These equations can be rearranged to be identical. They are different representations of the same line.

Students apply the theorems they proved in the previous activities to problems. First, students write an equation of a perpendicular line by recognizing that the slopes must have a product of -1. Next, students explain why two lines that are both perpendicular to the same line must be parallel.

Invite students to share their equations, and display this image.

Ask students what they notice about the equations for lines and (same slope) as well as the graphs (parallel lines). Invite students to explain whether this is always true. If students do not mention the possibility of lines coinciding, ask students what would happen if we graphed a line perpendicular to , passing through the point (the line would coincide with ).