##
* *Reflection: Connection to Prior Knowledge
Prove Slope Criteria for Parallel and Perpendicular lines - Section 2: Proof of the Slope Criterion for Perpendicular Lines

Originally, this lesson covered a lot more than it currently does. My school has 2hr blocks once per week, and so I had decided that on one of these block days, I was going to take students on a two-hour journey that eventually led to proving the slope criteria for perpendicular and parallel lines.

This journey began with an exploration of similarity relationships that arise when the altitude is drawn to the hypotenuse of a right triangle. This exploration, which included an introduction to the geometric mean is now its own independent lesson. After establishing the relationships that arise when the altitude is drawn to the hypotenuse of a right triangle, we then used these relationships to prove that the slopes of perpendicular lines are opposite reciprocals. Essentially, though, students were learning two full lessons, and a third on proving parallel lines have equal slopes, all in one sitting. It was a beautiful experience of coherence and it took a lot of hard work and framing on my part to pull it off, but was it the best approach? Looking back, it clearly wasn't.

So how did I get into this predicament of having to attempt such a feat. Well, when I originally planned the course I had an idea of how I would prove that perpendicular lines have opposite reciprocal slopes. The proof I had in mind had not involved the altitude to the hypotenuse or the geometric mean. It involved only the Pythagorean Theorem and systems of equations. But then, as I was thumbing through our textbook, I realized that one of the challenge problems involved this proof and the problem provided a hint: Draw the altitude to the hypotenuse. I liked the elegance of this proof compared to the proof I had intended to teach. However, I also realized that my students did not have the prior knowledge that would be needed for them to approach the proof in this way. So I needed to teach the prior knowledge and the actual lesson all at once.

Now that I know that I will approach the proof in this way, I have written the lesson on the altitude to the hypotenuse, and I teach it well in advance of this lesson. As a matter of fact, the learning from that lesson turned out to have multiple connections to other lessons. For example, the proof of the Pythagorean Theorem using similar triangles, and Construction Problems. So now, rather than students experiencing the coherence as a one-day event, they get the more powerful experience that the knowledge they gain from one lesson is actually powerful and allows them to access knowledge in subsequent lessons. This, I think, is more what the framers had in mind when they envisioned the shift to a more coherent curriculum.

*The importance of Chunking*

*Connection to Prior Knowledge: The importance of Chunking*

# Prove Slope Criteria for Parallel and Perpendicular lines

Lesson 1 of 7

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

#### Activating Prior Knowledge

*15 min*

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.

*expand content*

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.

*expand content*

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.

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#### Assessment

*15 min*

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.

#### Resources

*expand content*

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Prove Slope Criteria for Parallel and Perpendicular lines
- LESSON 2: Algebraic Proof of the Perpendicular Bisector Theorem using Coordinate Geometry
- LESSON 3: Loci and Analtyic Geometry
- LESSON 4: Equations of Circles
- LESSON 5: Proving the Medians in a Triangle Meet at a Point
- LESSON 6: Partitioning Segments in the Coordinate Plane
- LESSON 7: Prove Triangle Midsegment Theorem using Analytic Geometry