For this section, I write the following problems on the board as the warm-up for today's lesson:
This is in preparation for simplifying sinA/a, for example when sinA is already a fraction. When we get to the point in todays lesson when students have to simplify h/a divided by c, I want them to realize right away that it should be h/ac.
So after students complete the warm-up problems, I demonstrate them while pointing out the generalities of approaching problems like this where a fraction is divided by a whole number or single variable.
In this section, my goal is to make sure that students have the fundamental concepts and understandings they'll need in order to use the law of sines. These include the convention for naming sides and vertices of triangles, knowing what the law of sines says, understanding problems in which the law of sines has been applied, and understanding what information needs to be given in order solve for a certain part of a triangle using the law of sines.
Students will be working on Law of Sines Fundamentals. This handout is a teaching resource. It models several skills and concepts for students. Their task is to make sense of what is being presented. Toward that end, I step through the handout with the class as a whole, stopping to explain at key points. On each of the example problems, for instance, I stop to explain the thinking that goes into the solution and the steps that lead to the solution.
On the part where the students have to label the vertices of the triangle, I have each student fill it out on their own first. Next, I have each A-B pair play rock paper scissors to see who gets to explain why they placed the vertices where they placed them. The other member then echoes what was said and either agrees or disagrees. This continues until both partners agree.
Finally, when students are asked to provide the given information that would be required to find a particular part of the triangle using the law of sines. I have the students work alone at first, and then in pairs, taking turns explaining their answer and rationale.
In this section, I have my students working on Law of Sines Practice Problems. There are two problems that require them to use the law of sines to find a side length, two that require them to use the law of sines to find an angle measure, and two that require them to use the law of cosines. I like to throw in a couple of non-examples to make sure that students are thinking about the conditions for applying the law of sines. It's also a good opportunity to make sure they've retained what they learned about the law of cosines.
Typically, I would have students working in pairs on these problems and I would show detailed solutions on the document camera, one at a time, after the majority of the class has finished working on the problem whose solution I'm about to show.
In this section students will be working on Proving the Law of Sines for Acute Triangles.
The resource for this section is largely a teaching resource. Students will need to read carefully and understand what they are reading. If they do that, I feel they should be able to go through the handout self-directed. Erring on the side of caution, though, I don't leave students to work completely on their own.
My job, as I see it, is to regulate the flow as we proceed through the handout. When I want students to read independently, I have to quiet the class and make it mandatory for students to read a particular chunk of text on their own. If I want to make sure students have understood what they've read, I have to structure student talk that compels them to demonstrate their understanding. If I want students to fill in blanks on their own, I have to structure the time so that students work on their own and I have to walk around to make sure they are actually filling in the blanks.
The other part of my job is to paint the big picture for students by calling attention to the connections I want them to see. The main idea is that the entire handout is a coherent proof and all steps are necessary, intentional, and strategic. So as we're reading through the resource together, I want to make sure that students are not taking anything for granted. I'm continually posing the question "Why did we do that?" or, even better, "Why was it necessary for us to do that?"
By the end of this section, we've proven the law of sines, but only for acute triangles. If I am teaching an advanced class, we will be proving the law of sines for obtuse triangles in the coming sections. If not, then I will take this time to explain that the law of sines does apply to all triangles, but that the proof for obtuse triangles requires math that is "beyond the scope of this class". Of course, if there are any students who are still interested in the proof for obtuse triangles I will arrange a way for them to study the proof independently or else come outside of class time to get some instruction on it.
Up until this point, my students still have the idea of an angle as two rays with a common vertex. So this is a major conceptual shift. I start by providing some context with physics. I explain how in physics there are two types of motion: linear motion and angular motion. Linear motion, I explain, is motion in straight lines. Angular motion, on the other hand, is motion along a circular arc. So for example when a track athlete is running the 100 meter dash, we say they are running at a velocity of 10 meters per second, for example. But if an ice skater is spinning during a routine, we might say that she is rotating with an angular velocity of 700 degrees per second.
After this brief anecdote, I will give a definition of an angle measure as a distance along a circular arc. To bring this definition to life, I show students a Geometer's Sketchpad sketch that you can see in the following video.
In this section students will take it a step or two further in order to prove that the law of sines is true for obtuse triangles. But first I provide a basic introduction to the unit circle and I develop the identity sin (180-x) = sin x. Finally students prove the law of sines for obtuse angles.
All of that takes place in the Law of Sines for Obtuse Triangles
I do some direct teaching on the unit circle and trigonometry thereof. I also give a pretty direct explanation of the sin(180-x) = sin x relationship. I just want students to know this information and understand the concepts involved.
Once students get to the last two pages of the handout, they will be working independently to apply what they learned earlier and what they have learned in this section of the lesson.