Learning Laws of Sines and Cosines the Hard Way

Yesterday, I gave the honors kids a warm-up problem on the board (the hardest part was for them to read and follow all directions, apparently):

1. Draw an obtuse scalene triangle ABC, where A is the obtuse angle. (I announced out loud that the triangle should be "about two fistfuls" in size. "Don't be stingy!")
2. Measure sides AB and BC only. Label their lengths in the triangle.
3. Measure angle B only.
4. Draw a dashed line down from A to create height.
(I drew this on the board to show them how to line up their protractor to make sure the "height" is perpendicular to the base.)

At this point, I waited to make sure that everyone had an obtuse triangle with two sides and their included angle known.

Then, I gave oral instructions for everyone to fold the triangle back along the dashed line, so that only the right triangle with the known "included" angle is showing. They then proceeded to individually find all sides (no rounding!) and all angles (no rounding!) using basic trig. When they were done with that first right triangle, I had them open the fold back up to find all sides and angles on the other side, paying close attention to which pieces of information "carried over" to the other side...

In the end, this activity was GREAT because it was self-checking. After they did all of the calculations, verifying their results with a ruler and protractor was easy to do and very rewarding! And it built their confidence that their knowledge of right triangles could now be extended to analyze all types of triangles. (A precursor to Law of Sines.)

The rest of the class, they worked in groups on something that was kind of difficult for some of them (but very manageable for others).




Notice that all of these problems could easily be solved using Law of Sines, which I will introduce to them next week. But, I wanted them to do it the "long" way first -- cutting scalene triangles up into right triangles -- so that they can gain a real appreciation for A.) that they can do it on their own without some fancy formula, and B.) it's a lot faster with the fancy formula.

(The "fancy formula" Law of Sines comes from #2A on the second page, actually. It's just a matter of re-arranging those terms once you have the equation sin(A)*b = sin(B)*a, so I don't anticipate them feeling surprised when they see it next week.)

Today, my intention was to return to this and finish it in class with a discussion, but instead I gave the kids another Do Now similar to the one they did yesterday, but this time WAY HARDER (ie. it's the "hard way" of doing Law of Cosines.)

1. Draw an obtuse scalene triangle ABC, where A is the obtuse angle.
2. Measure all three sides of your triangle. Label their lengths in the triangle.
3. Draw a dashed line down from A to create height. Label the other endpoint D (directly below A).
4. Label BD as "x". You may NOT measure this length.

I had to give them heavy hints to setting up Pythagorean Theorem equations for both right triangles in order to solve for h^2 and to set both h^2 equations equal. In the end, they were able to get through the whole problem -- finding all sides and all angles -- and to verify their own answers with a ruler and a protractor, but I felt much less good about this. (The concept was just a step too complex for them to do all on their own, unfacilitated, the first time.) Even though I did little talking and they did most of the work, it still took about 35 minutes in each class -- a ridiculous amount of time for a Do Now!

But, I am hopeful! Next week will start off with a discussion of generally how to approach triangle analysis, and then a mixture of doing things the hard way (ie. still chopping scalene triangles up into right triangles) and then verifying their results using the Laws of Sines/Cosines "shortcuts." I am feeling really good about where they're at, because I can hear the gears turning in their minds as they push themselves to consider, "What do I know? What do I want to find? And how can I be strategic about getting there??" --which, to me, is the GREATEST part of math. :)