Prove Slope Criteria for Parallel and Perpendicular lines

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SWBAT use analytic geometry to prove that non-vertical parallel lines have equal slopes and perpendicular lines have opposite reciprocal slopes.

Big Idea

X-Games?? In this lesson, students are gettin' extreme on the slopes...of parallel and perpendicular lines that is.

Activating Prior Knowledge

15 minutes

When I put together an activating prior knowledge activity, I'm hoping that students will be able to complete it quickly on their own. What I definitely don't want is for a 10 or 15 minute activity to turn into a 30 minute activity. Therefore, I have to provide strict time constraints to keep things moving.

I give students Activating Prior Knowledge for Proving Slope Criteria. Then I give them 3 minutes to work on #1 through #3. If students remember the relationships, this is plenty of time. After the 3 minutes, I'll be giving the answers for students whose memories need jogging.

Note: Students will probably need guidance on how to complete the Therefore, _________ statement. So I give them a frame: Therefore, (blank)/(blank) = (blank)/(blank) and (blank)^2 = (blank)(blank) and blank= sqrt(blank times blank).

Next I give three minutes for students to work on #4. Again, this should be plenty of time for a student who knows what they are doing. When this time has elapsed, I demonstrate the problem and emphasize the concept of the point of intersection being the ordered pair that satisfies both equations, i.e., the solution to the system of the two equations.

Finally, I give students 3 minutes to work on #5 and #6 and then provide example responses for both.


Note: If I want to do some more intensive algebra review in this section, I guide students through Systems and Solution Sets. This activity involves systems of equations with no solution or infinitely many solutions. In it, I also introduce students to set notation if they've never seen it. Finally, we talk about what makes a rational expression undefined.

Proof of the Slope Criterion for Perpendicular Lines

20 minutes

Proving the Slope Criterion for Perpendicular Lines is a highly scaffolded resource. Developing students' reading comprehension is a long-term goal for me, so I do have students read quite a bit. First, I have students read through the resource once on their own, filling in the blanks as they go (in pencil). They should also be completing #1 through #4 at this time (I ask them not to start #5 at this time). Then I have students get together with their A-B partners to compare answers and verbally brainstorm how they will approach #5. Next, I review the answers with the entire class, explaining important concepts as I go.

Finally, it is time for students to work on their proofs in #5. By the time they set out to write their proofs, they have had time to think independently, they have collaborated with a partner, and they have heard my thorough explanations. Therefore, it is reasonable to expect that most, if not all, students would be able to do a decent job on the proofs. I notify students that I will be calling a few of them up to the document camera to present their proofs. This is the truth, but is also a way to motivate students to produce high-quality, legible work.

When students have had time to finish, I call 3 to 5 students up to present their work. As I hear the presentations, I give critical feedback and targeted praise as needed.



Proving the Slope Criterion for Parallel Lines

20 minutes

To begin this section, I give every student a copy of Proving the Slope Criterion for Parallel Lines. The first part of the handout asks students to fill in the blanks as they read to understand the basic logic involved in the proof. I have students read and fill in these blanks on their own first. Then I have each pair of A-B partners compare their answers and take turns summarizing the logical chain in their own words.

Once we get past this part, I call the class' attention to the front of the room so that I can explain the process of finding the equation for  x-coordinate of the intersection of two arbitrary linear functions. There are subtleties in the process that I want to make sure to explain and the mechanics of solving the equation for x may also present challenges for students. For these reasons, I am careful to explain each step, and the rationale behind it, very thoroughly.

Once I have explained a major concept or procedure, I will usually have students pair-share to make sure they understood the explanation. Once I feel that students have had adequate opportunity to access the information and concepts required to write the proof, I have them work with their A-B partners to rehearse the key steps of the proof.

When students have finished rehearsing, I have them turn their papers over and begin writing the proof. As in the last section, when students are finished writing, I will either call a few students up to the front to share what they have written or I will have all students exchange papers with a peer and then give each other feedback on their proofs. In the latter case, I would follow up with my own version of the proof or a student exemplar.


15 minutes

I have students take all of the lesson resources from this lesson home with them so that they can study them. I tell them that they will have a quiz that will assess their understanding of the lesson. So when they come in on the day of the quiz, I hand them a copy of Quiz_Proving Slope Criteria and they have 15 minutes alone with their thoughts to complete it.