12.1: Parallel and Perpendicular (5 minutes)
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.
The image shows line \(n\).
- Write an equation for the line that is perpendicular to \(n\) and whose \(y\)-intercept is \((0,5)\). Graph this line.
- Write an equation for the line that is parallel to \(n\) and that passes through the point \((3,1)\). Graph this line.
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.
12.2: Info Gap: Lines (20 minutes)
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.
Design Principle(s): Cultivate Conversation
Supports accessibility for: Memory; Organization
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
- Silently read the information on your card.
- Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
- Before telling your partner the information, ask, “Why do you need to know (that piece of information)?”
- Read the problem card, and solve the problem independently.
- Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
- Silently read your card and think about what information you need to answer the question.
- Ask your partner for the specific information that you need.
- Explain to your partner how you are using the information to solve the problem.
- When you have enough information, share the problem card with your partner, and solve the problem independently.
- Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
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.
12.3: Three Lines (10 minutes)
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.
Design Principle(s): Optimize output (for justification); Cultivate conversation
- Line \(\ell\) is represented by the equation \(y=\frac23x+3\). Write an equation of the line perpendicular to \(\ell\), passing through \((\text-6,4)\). Call this line \(p\).
- Write an equation of the line perpendicular to \(p\), passing through \((3,\text-2)\). Call this line \(n\).
- What do you notice about lines \(\ell\) and \(n\)? Does this always happen? Show or explain your answer.
Are you ready for more?
Prove that the line \(Ax+By=C\) is always perpendicular to the line that passes through \((A,B)\) and the origin.
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 \(\ell\)).
Display this image for all to see.
Ask students to think of as many methods as possible to create new lines that are either parallel or perpendicular to line \(\ell\). Encourage students to think back to earlier units. Ask, “How can we use algebra? How about transformations? What about using a compass and straightedge?” Here are some responses students may give.
- Find the slope of line \(\ell\) and create another line with the same slope.
- Translate line \(\ell\).
- Rotate line \(\ell\) 180 degrees using the origin (or any point not on the line) as a center.
- Construct a line perpendicular to line \(\ell\), then construct another line perpendicular to the new line.
- Find the slope of line \(\ell\) and create another line with a slope that is the opposite reciprocal of this slope.
- Rotate line \(\ell\) 90 degrees using any point as a center.
- Choose two points on the line. Then, use constructions to create the perpendicular bisector of the segment contained between the two points.
12.4: Cool-down - Describe the Line (5 minutes)
Student Lesson Summary
We can use the concepts of parallel and perpendicular lines to write equations of lines. The image shows line \(\ell\).
Suppose \(n\) is the image of \(\ell\) when it is rotated 90 degrees using \((\text-1, 5)\) as a center. What is an equation of line \(n\)?
The point \((\text-1, 5)\) is on line \(\ell\). The center of rotation does not move when a figure is rotated, so \((\text-1, 5)\) will also be on the image, line \(n\). Because line \(\ell\) was rotated 90 degrees, lines \(\ell\) and \(n\) are perpendicular. Their slopes must be opposite reciprocals. The slope of line \(\ell\) is -3, so the slope of \(n\) is \(\frac13\). Now substitute the slope \(\frac13\) and the point \((\text-1,5)\) into point-slope form to get \(y-5=\frac13 (x-(\text-1))\).