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?

ANSWER = Geogebra  

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.

Fun With Grow Objects

I would like to dedicate this blog to today's snow day.  Without you, this posting would not be possible. (Or definitely not at likely.) This is the story of a polar bear, fruitcake, snowflake, Christmas tree, candy cane, penguin, snowman, Santa, gingerbread man and a Geometry class.

At the beginning of December, I went into Michael's to purchase what I call "grow objects".  (Very cheap - only $1 each!) You place them in water and they expand in size.  The claim on the package is they grow up to 600%.  Perfect!  We were studying similar figures in Geometry and students easily agreed that the enlarged objects should be similar to the original objects.  This mini project was perfect for sooooo much mathematics.  Prior to putting the objects in water, we considered the following questions:

1. What does it mean to grow 600%?  What is the scale factor?
2. Is the growth rate linear, exponential or something else? What equation could be used to model the growth?
3. Do all the objects grow at the same rate? When does each object grow the fastest?

Here you can see the objects at the end of 10 days of growth.

Due to snow days, midterm exams and Christmas break, my Geometry students have only investigated the answer to the first question.  What does 600% growth mean?  Does it mean all dimensions will be 6 times larger?  To answer this question, we considered a cube with a side of length 1 and then a cube with a side of length 6.  Students quickly saw that if all dimensions were 6 times bigger the volume was 216 times bigger!  

So, what would the side of the cube be if the volume was 6 times larger than the original?  With a little discussion, we decided it would be the cube root of 6 or about 1.8 units.

Students then reasoned that this would mean that the objects would grow to a little less than double for each of the dimensions.  So, how did the objects do?

All objects more than doubled in each of their dimensions!  In fact, for most of the objects the actual growth was between 900% and 1100%.  Maybe the manufacturers should relabel the packaging.  After all, 900% and 1100% sounds much more impressive than only 600%.

Unfortunately, our unit on coordinate geometry and modeling doesn't happen until April and questions 2 and 3 don't lend themselves to the study of the Pythagorean theorem, our current unit.  However, in researching for this blog, I found out that I could buy some brains, lions, spiders, geeks, castles, money and unicorns for growing at onlinesciencemall. The students had so much fun with this that we will definitely be doing this again.

However, you are probably wondering the answers to questions 2 and 3 and I was, too. So, I graphed two sets of data from my students (Christmas tree and polar bear) with desmos (a wonderful FREE online graphing calculator) and found some equations to fit the data. It should make sense that a cube root function works best since the growth is impacting the volume of the object and students were only plotting days against one of the three dimensions of the object. So that answers question 2. (Click here to see/download this desmos file that I placed on google drive.)

But what about question 3?  With Desmos we can easily plot the derivative of each function.  Notice that they are very similar to each other. This makes sense since f(x) and g(x) have the same basic parent function, the cube root function.  Also, from the derivative we can see that the fastest growth occurs between days 0 and 2, where the derivative has the largest positive value.  The asymptotic nature of the derivative also makes sense since the growth of the objects slowed down considerably by day 5.

Who knew that polar bears, fruitcakes, snowflakes, Christmas trees, candy canes, penguins, snowmen, Santas, and gingerbread men could help students learn so much about similarity, modeling and rates of change?