## Reflection: Backwards Planning Law of Cosines - Section 1: Proving the Pythagorean Trig Identity

When I first taught this lesson, I had my own idea of how to derive the Law of Cosines. Of course the pathway I had chosen was not the only pathway. Unfortunately, though, I had limited my thinking to the pathway I had chosen.  As a result, I designed the lesson in a way that funneled students toward this particular pathway.

As things tend to go, there are always independent thinkers who have their own ideas and like doing things in different ways. During this lesson, I had a number of students who chose to involve the sine ratio rather than focusing solely on the cosine ratio. Initially, I discouraged this. Frankly, I discouraged it because these students were pursuing a pathway I hadn't anticipated, and I was uncertain about where it would lead. This reasoning was of course absurd and completely counter the real goal of the lesson: getting students to use math creatively to derive a new formula. I was basically discouraging creativity.  Embarrassing as it is in hindsight, I said to these students something senseless like, "Since this is the Law of Cosines, let's focus on cosine and not involve sine." What I was really expressing, like I said, was "You're heading down a path I hadn't anticipated and I'm not certain about where it leads."

Later on, after trying this pathway myself, I realized that it was a viable pathway, but only if students knew the identity sin^2(x) + cos^2(x) = 1.

To account for this, I revised the lesson to include this section introducing and proving that sin^2(x) + cos^2(x) = 1.

This taught me the importance of anticipating as many pathways as I can when dealing with an open-ended problem such as this. It also taught me that no matter how prepared I am, it is always possible that students will explore pathways that I have not anticipated. I have to be comfortable with this, encourage it, and be willing to learn from it.

Preparing for Multiple Pathways
Backwards Planning: Preparing for Multiple Pathways

# Law of Cosines

Unit 7: Right Triangles and Trigonometry
Lesson 5 of 6

## Big Idea: Deriver's...start your engines!...In this lesson, we'll be doing some heavy duty mathematics to derive the Law of Cosines.

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135 minutes

### Anthony Carruthers

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