And one of my favorite things to do is to ask for a volunteer to work/answer a problem. If they get it right, they get to pick the classmate to answer the next one. And so on around the room. They like this a lot; I see a lot of paybacks where A called on B yesterday so B calls on A today, but it’s kept friendly. So I like the class participation part of the grade, but I don’t generally leave it up to the students; I’ll engage them and make them earn that 10%. With the way the electronic gradebook works, I don’t enter the class participation grade until the end of the quarter, so a borderline grade will often get a small boost at the end, which is nice.
But there are some major themes running through precalculus which give us a few categories to classify the disparate subjects into. I talk about precalculus as being a little bit Algebra III (into which I group matrices), a little bit Trigonometry (which I also call Geometry’s big brother), a little bit Complex Numbers (into which I group polar graphing and vectors), and a little bit Statistics/Probability. So those are the four major themes I’d identify.
When we’re done, I stress to them that from the quadrant 1 sin values we can get the quadrant 1 cos values. From there we can get the rest of the quadrant 1 trig values. Knowing how the reference angles work and the positive/negative pattern, we can get the entire rest of the table. So all we need are 5 little values to get the entire table. And I discovered a few years ago there’s even a pattern there:
deg 
rad 
sin 
cos 
tan 
cot 
sec 
csc 








And this is great for strong math students, but some people think differently. For limits on rational expressions of x as x approaches infinity, the best way to convey what happens may be to just use really, really big values of x and plug the expression into a calculator. This type of approach is called numerical. For the example above, you can show how the x^2 value gets smaller and then larger again, and how 3x+4 just gets larger.
A graphical solution would involve graphing the lefthand and righthand expressions and seeing where they cross. There are downsides, of course; you need to know that your graphing window includes all of the solutions you’re looking for, and while that point looks like (4,16), is it exactly (4,16)?
But while I used to look at the graphing calculators as a crutch to critical thinking skills, I now see it as a useful tool in allowing the student to use numerical and graphical methods to solve problems. And I feel this is beneficial because each method can strengthen the comprehension of the other methods. For a student who thinks visually, the use of the graphing calculator may aid in his understanding of why the analytical method works, and so on. So while I’m a little late in incorporating the graphing calculator into the precalculus curriculum, we’re doing it this year.