Lesson 12

Students practice writing equations for lines parallel and perpendicular to a given line. Monitor for students who use slope-intercept form for the first question and for those who use point-slope form.

If students calculate an incorrect value for the slope of a line perpendicular to \(n\), 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.

If possible, invite one student who used point-slope form and another who used slope-intercept form to share their solutions. If no student used slope-intercept form, ask students if we have enough information to use the form \(y=mx+b\). Once both equations are displayed, challenge students to rearrange one of the equations to show they’re equivalent.

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

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need 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 ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

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.

Tell students they will continue to work with parallel and perpendicular lines. Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

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. 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 \((\text- 0.8, 4.4)\), their equation will look like \(y-4.4=2(x-(\text-0.8))\). If students used the \(x\)-intercept, \((0,\text -3)\), their equation will look like \(y=2(x+3)\). 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 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 for all to see.

Ask students what they notice about the equations for lines \(\ell\) and \(n\) (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 \(p\), passing through the point \((\text-3,1)\) (the line would coincide with \(\ell\)).