## Friday, January 17, 2014

### Teaching Trig Ratios in Geometry

Although I have probably taught trig ratios (sin, cos and tangent) 2 or 3 times each year for about 15 years, I was never really satisfied with how it turned out.  Yes, students could parrot the definitions and find sides and angles until the cows came home, but did they really understand why the sine ratio was always a certain number for a specific angle?  I was doubtful that the big picture was being understood...until today!  OK...I am still not satified with the entire lesson, but I think it was better.

As much as possible, I want students to construct their own knowledge/understanding of concepts with my guidance at the side.  In the past, I would have them draw right triangles that were similar, measure segments and calculate various ratios. (See handout below.)  But guess what the problem was.  Yes, that is right.  Inaccurate measurements led to the ratios not being equal and me saying things like "Well, the ratio 0.75 is pretty close to 0.76.  So, close that if you had measured accurately, I bet they would be equal."

Bleh!!!

The world would be a utopia if the students could measure accurately with 25-cent rulers and 25-cent protractors.  But they haven't and can't.  How could I get more accurate measurements?

I was going to create a worksheet for them to do, but didn't quite have enough time.  (Both enough time in class and enough time to create the worksheet.) So, then I decided to create a screencast; it is uploaded to my youtube channel and I was planning on showing the 6 minute screencast in class. But since I really wanted students to experience the discovery for themselves, I led them through the activity as a whole class with each student working on his or her own laptop.

As long as students did exactly as I did and clicked in the exact same order, we were ok.  If they clicked in a different order, then we had different outcomes; their segment d was the hypotenuse and my segment d was the leg.  Luckily, only one or two students clicked in a different order and they figured out how to make adjustments for what they had on their own computer.

Students in my one class were initially surprised that we all got the same values, and when I asked them why this made sense, they understood!  Students chimed in that all the ratios must me equal,  because the triangles we had on our screens were all similar to each other by Angle-Angle similarity!

I think some students didn't find this too amazing, because as a class we decided what angle we would use and we all used the same angle.  Their feeling was "So, what?" Next year I still plan to lead the students through this activity rather than have them follow a set of typed directions.  However, I would have the data collected by table. I would have each table group choose what acute angles their right triangle would have.  This means there would be 5 sets of similar right triangles and they would see that it is not just one specific set of triangles where the sin ratio is the same, but the sin ratio is the same for any one given angle.

1. Nice! I like Geogebra, but I do the activity with each row or group being assigned a different angle. We then generate a trig table for the angles (usually multiples of 10). In this way each row can see that their triangles are different (I am fine with the error of the 25c rulers) and now we have some other interesting patterns--the "1st and 2nd columns" are reversed and with nudging from me they see the third column is the quotient of the first 2.

I actually like that the answers are close.

Now I introduce the formal terminology and ask them to use the sin, cos, and tan buttons to see which column appears to related to which buttons.

I like the incorporation of Geogebra here as well. With better access to computers next year, I hope to make Geogebra a regular part of the learning process.

2. As I read this post I thought, "Wow, this is exactly the process I went through!" I made an applet for Geogebra that kids can use and a worksheet to go with it: http://drawingonmath.blogspot.com/2012/06/trig-intro-applet.html

3. This blog post & Tina C's applet (linked above) motivated me to create a Geogebra lesson exploring trigonometry. Thanks for the idea of using Geogebra for this purpose!
Here is the worksheet:
4. 