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
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Prove Slope Criteria for Parallel and Perpendicular lines

Unit 9: Analytic Geometry
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.

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

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perpendicular slopes
 
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