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
Loading resource...

Law of Cosines

Unit 7: Right Triangles and Trigonometry
Lesson 5 of 6

Objective: SWBAT derive the law of cosines formula and use it to solve problems

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

  Print Lesson
Add this lesson to your favorites
law book
Similar Lessons
Triangles That Are Wrong Because They Are Not Right
12th Grade Math » Additional Trigonometry Topics
Big Idea: Investigate how to find side lengths and angle measures for non-right triangles.
Troy, MI
Environment: Suburban
Tim  Marley
Final Exam Review Stations (Day 1 of 3)
12th Grade Math » Review
Big Idea: Students review by working through various stations at their own pace and receive immediate feedback on their work.
Phoenix, AZ
Environment: Urban
Tiffany Dawdy
Ambiguous Case Day 1 of 2
12th Grade Math » Solving Problems Involving Triangles
Big Idea: After analyzing the ambiguous case for oblique triangles students will determine the number of possible solutions and find solutions when possible.
Independence, MO
Environment: Suburban
Katharine Sparks
Something went wrong. See details for more info
Nothing to upload