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* *Reflection: Vertical Alignment
Prove Slope Criteria for Parallel and Perpendicular lines - Section 1: Activating Prior Knowledge

At this point in the year, I am realizing that we've reached the home stretch of the geometry course. Students are beginning to fill out their course requests for next year. Many of them have requested honors algebra 2.

It dawns on me that, even though I am teaching Geometry, I have a responsibility to make sure that my students are prepared to succeed in Algebra 2. This unit could not have come at a better time. In this lesson, we are hitting on systems of equations and rational functions: two topics of great importance in higher algebra courses.

I am being very intentional about raising the level of sophistication when I introduce or discuss these algebra topics. For example, I really enjoyed introducing set notation and discussing what the solution set of a linear equation in two variables really encompasses. It was nice to see students understanding something that they thought they had known from Algebra 1 on a much deeper level. In the same way, it was nice to discuss why it makes sense that dividing by zero is undefined.

So going forward, I realize that this unit on Analytic Geometry is a unit that I really want to treat as a Rites of Passage experience for students who are moving on to higher, more formalized levels of algebra after they leave my class. This will remind me to keep the level of sophistication and precision high and also to be explicit with students in telling them that this level of rigor will be the norm in their next course so start preparing now.

*Preparing for higher algebra*

*Vertical Alignment: Preparing for higher algebra*

# 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.

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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.

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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

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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