Laws of Sines Derivation and the Ambiguous Case

I really wanted to make these lessons more meaningful, so I started off with a 3 act math task from the website http://www.101qs.com. This one addressing finding how many pieces of sod cover the triangular garden shown below:

 This led to some good questions from the students that made them want to learn how to solve oblique, or non-right triangles.

First, students derived the Law of Sines

Then they practiced a bit. Then they derived the area formula for a triangle using the Law of Sines:

After practicing that a bit, they were able to solve the sod problem...though they got 18 pieces and the video got 20. Hmmmm.

Anyway, the next day we tackled the Ambiguous Case (SSA) for the Law of Sines. We talked about why there could be an ambiguous case for sine and not cosine, i.e., there are two angles less than 180, one that is acute and one that is obtuse for every sinx = a number between 0 and 1. For cosine, if cosx = negative number, the angle would be obtuse and if positive, the angle would be acute. 

I told them that there were three possibilities with SSA-two triangles, one, or none. They were reminded of the discriminant at that point! I had each case on a separate piece of paper with scissors, pink paper, rulers, and a glue stick. For each case, they had to draw line segment AB about halfway down the paper, knowing that they did not know it's length (this is also side "c".) They may need to draw it longer or erase from it. But they had to measure the given angle A and cut length "b" from the pink paper. Then they had to cut length "a" to try and have it meet side c (AB). Here's what they did for the first one.

They measured the sides and angles that were missing and then solved by Law of Sines and saw that the answers were very close...and that there was only one possibility for this triangle. When they solved for B, we talked about how B' = 180 - B was the other solution that gave the same sine value. However angle B' + angle A > 180, so there would only be one triangle.

We did this again for case 2, and they saw that no triangle exists, as shown below. 

We proved this mathematically, which was pretty cool. I should point out that my sides were a bit long, and next year I will make them smaller...however, students knew that the ratio of the sides just had to be in proportion, so they divided by two in some cases to get them to fit on their paper. 

We all had enough of the arts and crafts and for the last case, two solutions, we drew it by hand. It was very cool to see the students grasp the concept of side "a" swinging in to make an oblique angle, and that the two angles, I call them B and B', one acute and one obtuse, both added to 180 degrees. They also saw that angle B' + angle A was less than 180 degrees, so they had room for a third triangle, and therefore there were two possibilities algebraically. Or shall we say, trigonometrically.

They solved for the missing sides of both triangles, and practiced one of each type after that. I felt that they understood the ambiguous case much better.

Tomorrow, we will derive the Law of Cosines and after that, Heron's or Hero's formula. 

Here are my notes for Law of Sines and Area derivation and practice, Law of Sines Ambiguous Case.

Puzzle Day: Human Jumping Puzzle, Riddle of the Tiled Hearth, and 8 Queens

Today was a fun day in problem-solving. I have been wanting to do the frog jumping puzzle Shaun Carter blogged about since his first post last year. He also made this interactive puzzle shown below:

We did the human jumping puzzle by taking 7 chairs and putting them in a row and making two teams: 3 blue and 3 reds. With AP exams, we happened to have exactly 6 students in class, so it worked perfectly. I recommend that you have students wear a colorful tag because even though they knew who was on each team, I could not keep it straight. The rules are above, but in terms of humans,

• Blue team can only move to their left and red team can only move to their right.
• You can move forward one chair if it is empty and next to you. 
• You can move forward two chairs if you "jump" over someone
• The puzzle is solved when all of the blue team moves to where the red team was and vice versa (order does not matter.

The goal was for my students to do this in as few moves as possible. They did it about 3 times (they are a bright bunch) until they got it right: 15. I wish I took pictures!

We then took away chairs and did it again with 2 blue and 2 reds and also 1 blue and one red and found the pattern to the number of moves it takes, which you can also see on Shaun's post. My students recognized the pattern as perfect squares - 1, starting with (n+1), where n is the number of half the counters (so, only the blue counters, for example.) 

This reminded me of another puzzle, which I read about in this (parentheses), [brackets], and {braces} blog, called The Riddle of the Tiled Hearth. There is a great story that goes along, and we used unifix cubes and played with this for a while. In a nutshell, in a 4x4 grid, you have to place as many of four different tiles (or colors) in the grid without any two colors being in the same row, column, or diagonal...kind of like a type of Sudoku...using the least amount of blank spaces as possible. This took quite some time, and it was interesting how the kids really thought they could do it without having any blank tiles (you can't.) We talked about how tiles had to be "knight moves" away from each other, from chess, so that they were not in the same row, column, or diagonal.

From ppbmath.weebly.com

Then THIS reminded me of ANOTHER puzzle: The Puzzle of the Eight Queens. They loved this one, too...and I did not have an 8x8 sheet ready, but next year I will. The object is to place 8 queens on a chessboard so that none can attack one another...again, none can be in the same row, column, or diagonal. Students could adapt this from the last puzzle by using 8 cubes of one color. The website I linked is cool because students can check their work if you project it on the board and have them tell you their answers.

These were all really fun, and I joked with the kids that they could take the rest of the day off in terms of thinking in the rest of their classes...our heads hurt! Not all of them could solve every problem, but it certainly gave them background info to solve a similar problem in the future.