I am teaching Geometry for the first time this year. Which means I get to teach right angle trig from scratch! In the past I have taught trig (circular) functions, but never the more basic bits.
I was supposed to teach "triangles" which they already knew and had for some time (my school seems to have accepted that math will be boring and repetitive). So rather than belabor the point, I did a quick review of terms that led to the idea of similarity. I had them apply the idea of similarity to bigger things, like the school building... or the playground equipment... or the local mountain top.
Within a class period most if not all understood the idea that the ratio of two sides of a triangle can only be changed if the angle changes, in fact when I brought it up half the class gave me that look of "duh mister, we figured that out." This was all done with no mention of sine or cosine.
With the real learning done, I started into a bit of formalization. With a short back and forth discussion that required students to draw a few more triangles to convince themselves of a the details we had the 3 main trig functions on the board in just a few minutes, but still unlabeled.
With nothing more to teach I finally labeled the functions... I gave them some typical problems to practice with, only one of which I did on the board. Then they were off and running. I felt so close to a masterful introduction of trig. So very close.
Next on my list was inverse trig functions...
I posed the question (with a drawing on the board), given two sides of a right triangle how do we find the angle? I figured there would be lots of drawing and struggle to draw the correct triangle in order to measure the angle. So close, I could almost taste it. Somewhere somehow during this time someone shared that they could use the calculator to find the angle, i.e. inverse trig functions. They don't understand inverse functions, but they have buttons to push...
I was so close. Sometimes I despise the power that buttons seem to have over students.
Should I have given them a table of trig functions instead of allowing them to use their calculators? That seems so contrived, so fake, so false. How do I avoid this next year?