Later in this lesson when we derive the Law of Cosines, depending on the route they take, some students may find themselves in a position where they need the identity sin^2(x)+cos^2(x)=1. So in this section I introduce this identity and guide students through the process of proving that it is true.
Aside from preparing for the lesson, there is also some cool math in here that will have longevity as students go on to higher math courses. For example, the concept of identities and how we prove them.
I give each student Proving Pythagorean Trig Identity. We read the text together. I stop to clarify, reiterate, or elaborate as I see the need. I have students work in pairs to complete #1 through #4. Then I show the correct responses to these so that students have the right information. Finally I give students 5 minutes or so to work on #5 before I show the correct response to that. If I have more time to allocate, I will have students come to the front of the class and present their proofs under the document camera.
In the lesson previous to this one, students derived the SAS triangle area formula. In this section of the lesson, I use that derivation process as an example to discuss more generally how we often go about deriving formulas.
I explain that in the previous lesson we wanted the area of a triangle in terms of two sides and the included angle. My explanation hits the following bullet points:
This is a good opportunity to review what we learned in the previous lesson, but primarily it is a way for me to train students in type of thinking they'll need to do to derive the law of cosines formula.
To begin this section, I remind students that the learning goal is to derive the law of cosines formula. But first, I explain to them, we'll be facing some specific problems that could be solved using the law of cosines. Once we've gotten experience solving specific problem of this type, we'll then move on to deriving the general formula for solving this type of problem.
I tell my students that we're going to pretend that the law of cosines formula does not yet exist. I explain that formulas usually arise after people repeat a process without the formula until they realize the process could be made more efficient, or even automated, by creating a formula (MP8).
I will have students working on Specific Case of Law of Cosines Problem in groups of four. Each student in a group gets a different version of a problem in which two side lengths and the included angle measure are given. Giving the students four different, but similar, problems makes it possible for them to help each other, but it also pushes them to focus on process rather than specific answers. This is the perspective they'll need when we start to derive.
As I walk around, the first obstacle for students tends to be naming the partial side lengths that result from the altitude being drawn. We've encountered this situation before, where c is split into x and c-x, so I try to remind students of their experiences with this.
Once students are finished with their individual problems, they work together on the group sheet. The last part of the group sheet includes work on expanding binomials and solving radical equations. These skills will definitely come into play during the lesson.
To begin this section of the lesson, I give every student Deriving Law of Cosines_v1. I want to make sure that all students get on the right track at the beginning because they will have to work hard enough already to stay on track once they get there.
We read the first paragraph of text together. I ask pair-share questions to make sure that students are understanding what has been read and can relate it to the diagram.
I take some time to explain the meaning of mindful algebraic manipulation (as opposed to knee-jerk, or mindless, algebraic manipulation). I provide the following example to illustrate:
I say to my students, "Suppose we have the equation 3(15x-9)=63...what would be our first step in solving this equation?" And then I just wait for the instinctive chorus of students blurting out "DISTRIBUTE!!" (It seems that somewhere along the way, most students I've encountered have taken an oath to distribute whenever possible.) So after that dies down, if no one else suggests it, I say, "Students, what if...instead of distributing, we mindfully chose to do something else that would make things easier for us. What if we divided each side by 3? Then we could see that 15x-9 = 21, 15x = 30, and x = 2. That's easier than distributing and still having to deal with 45x-27 = 63"
Then my next question is, "So does that mean we should always divide first in this situation?...Not if we're being mindful. If we're being mindful, we'll check to see if the expression on the other side of the equation is divisible by the number by which we want to divide. If not, then distributing is the better pathway. This illustrates mindful manipulation."
Next, we go through the cycle of Think-Pair-Share-Teacher demonstrates on document camera for each of the items 1-5. In this phase, I'm trying to make sure that students understand the problem. This mostly means that students are going back and re-reading the text in relation to the diagram while considering the prompts in 1-5.
When this phase is complete, I have students move close to their A-B partner. I then give instructions for each pair to take turns coming up with the 5 equations (#6 and #7) one at a time, then choosing a starting equation (#8), and finally deciding which variables need to be cleared/introduced from/to the starting equation in order to get to the final formula (#9)
When the student pairs have had enough time to complete 6-9, I call the students' attention to the front of the class. I quickly show the 5 equations the students should have come up with and I explain the rationale for the choice of the starting equation. Finally, I give a careful explanation of #9, as this is a place where some students get confused.
I explain that our starting state is y^2+(b-x)^2=a^2 and that our end goal is an equation that will give us the value of a (our output variable from page 1) as long as we have values for b, c, and measure of angle A (our input variables from page 1).
Next I ask my students to notice that measure of angle A is part of our end goal, but not part of our starting state. I also ask them to notice that x and y are not part of our end goal, but they are part of our starting state. Our task then is to get x and y out and get measure of angle A in. I explain that they will be using mindful algebraic manipulation (and maybe some trial and error) to find a path from our starting state to our end goal. I tell them that they have a nice blank white canvas (#10) on which to experiment and "play around" (so don't try to be perfect...just try things and find out what accomplishes the goal). Then it's time for them to get to work.
After teaching this lesson, I've come to realize that some students get stuck and don't know how to get unstuck. These students tend to give up on the problem. To address this I've learned that I need to periodically come up to the board and write down a key step or hint that will give these students access to the problem. See the reflection in this section for an example of this.
Other than that, there are at least two solution paths that are valid that I need to be aware of as I'm checking in with students. Students who decide to incorporate the sine ratio will need to use the Pythagorean identity to simplify their formula (after all, it's not called the law of sines and cosines). I also have to be aware of common student errors (e.g., (x-y)^2 = x^2 - y^2 and sqrt(x^2 - y^2) = x - y.
As students start to get the correct formula and I confirm it, proud celebrations periodically erupt around the room and other students tend to gravitate to these centers in order to get some insight. I don't mind this at all and I allow it to occur naturally. Once a student is ready to move on to #11, I remind them that their role has now changed from solver to communicator. I explain that they have creative control over their work presentation and advise them to exercise that control conscientiously.
By the way, as an extension for advanced students, as an alternate version for a different class period (students tend to talk at lunch), or as a follow-up activity to this activity, I have Deriving Law of Cosines_v2. By simply changing the designation of x to the other segment created by the altitude to segment AC, the problem becomes qualitatively very different. This is a good opportunity for students to experience how choices that seem insignificant can actually make a big difference, and how that teaches to be mindful about our choices.
As a follow-up to this lesson, I give students Apply Law of Cosines to complete at home. Students have learned everything they need to know in order to complete the problems (in this lesson or in previous lessons) so I don't feel it's unfair to have them work on it independently.
At the same time, I want to make sure that they know how to do the problems so at the next class meeting, I will demonstrate these problems thoroughly. Otherwise, I will post detailed solutions on the internet.
Depending on time, and the level of my students, I may also teach students to solve problems involving Resultant Vectors. Otherwise, I use this an extension for advanced students.