Friday, April 6, 2012

VFC: Intro to Trigonometry

Here is a Virtual Filing Cabinet for an introduction to unit circle trigonometry. The goal of this unit is roughly to get students conversant in the sine, cosine and tangent functions. Following this unit in my class is a full unit on graphing the trigonometric functions.

What am I missing here? Point me to your favorite trigonometry resources in the comments.

[Last Updated: 4/5/2012]

The Hard Parts

Part of what makes this unit challenging for me is that students may or may not come in knowing the right triangle trigonometry definitions of sine and cosine. Some teachers like to start with the right triangle definitions and expand them to all real numbers. Because I know that I can't rely on my kids remembering the old definitions, my solution is typically to offer a fresh start with the circular definitions of the trig functions, and then later tie them back in to right triangle definitions.

What I like about trigonometry is that so much of it can be connected to one central model -- the unit circle. This is empowering. But kids had better know their unit circle stuff, and so kids really need a firm grasp of the unit circle by the end of the unit.


This song goes first. It's fun:

Kate and Riley both have good starting points for the beginning of this unit, but my understanding is that they start with the right triangle definitions of sin, cos, and tan. I don't like to start there for two reasons. First, because my students usually don't remember those definitions. Second, because I'd rather give them a clean foundation and connect it to the right triangle definitions later.

I detailed my preferred starting point in a post:.

In general, I like the worksheets and implied approach of the eMathematics textbook. I use their worksheets on angles and rotation terminology after I've reviewed the equations of circles with my classes. You can find them at the link below, though everything from their Lesson #6 and on I consider part of my next unit, on graphing the trig functions.

I don't know where I first saw that the tangent function is the slope of a radius in the unit circle. I never learned it that way, but it makes so much sense to me and my kids than just defining tan directly in terms of sin and cos.

Here's a full course through problem sets. I haven't looked through it carefully yet, but it looks good:

Oh yeah, radians. And if you're in NY, arc length. I spend a day converting unit with my kids, and radians <---> degrees is an application of converting units. Sometimes I use this not-so-great resource that I made that just contains some goofy units of measurement to shake kids out of their comfort zone.

I'm torn as far as converting units is concerned. On the one hand, there's a procedure for converting units which can help you convert when you don't have an intuition for the relative sizes of the units. At the same time, gaining intuitions about the relative sizes of units is helpful. So my approach with radians/degrees is to both things. First I try to move them out of their zone of intuitions with an activity, and then they calculate "about how many degrees are in a radian?" to give them intuitions. I've also found that kids in my classes get confused about radians. They say things like "The unit circle has 2 radians" as opposed to "2 pi radians." What I've learned to do is ask them to estimate the number of radians as around 6, and ask them to make back-of-the-envelope calculation with that number, and then use pi for the more precise calculations.

Also, I teach arch length as a proportion problem. Meaning, what percentage of the circle is 2 radians? If the radius is 5, what will the circumference of the full circle be? What percentage of the circumference are we looking for? I know that's probably a no-brainer for a lot of you, but I had trouble coming to that approach.

Intro to Trig

Round these internet parts, a lot of people are fans of Kate and Riley's intro to unit circle trigonometry. Their introduction is designed to get students seeing that any point on the edge of the unit circle is (cosA, sinA). Mine is too, but I go about is slightly differently, and it seems to work.

In short: My first goal when teaching trigonometry is to change the way that my students see circles. When they see a circle, I want them to see it as full of right triangles. Then we spend a day talking about special right triangles inside circles.

We start with drawing a lot of right triangles:

And eventually kids end up with a picture that looks like this:

Then I ask them questions like, "Let's say we're on the edge of the circle and the x-coordinate is 4.3. What's the y-coordinate?" And from there it's a pretty quick jump to the equation of a circle. Then we move circles around, use the Pythagorean Theorem on then. By the time we're done, we define the unit circle and offer its equation.

That's great, but what's really important about that to me is that we've gotten in the habit of seeing circles as made up of right triangles. That gets us off to the races for developing the definitions of sin and cos. We lose some of our momentum when we spend a day getting used to rotation terminology, and we need to review the special right triangles. I'd like to tighten up my transition from the opening activity to this one:

Intro to Trig P5 Day Four

Intro to Trig Day 5 Worksheet

Tan Function1

Honestly, I can see myself combining this with Kate or Riley's intro activity. The purpose of all of these introductions is the same -- immersing students in a single, concrete conceptual model that students can return back to throughout trigonometry. My introduction is just my attempt to get right triangles in the mix from the very beginning.

Here's the powerpoint file from the beginning of the unit on equations of circles:

Equations of Circles Opening Activity

Monday, April 2, 2012

Sharing and organizing stuff

One of my ongoing obsessions is thinking about better ways of organizing the output of teachers on the internet. I've found efforts such as Sam Shah's Virtual Filing Cabinet  incredibly helpful, and efforts such as BetterLesson lacking.

What I've come around to is the idea that individual visions are far more helpful to teachers than crowd-sourced material.

Crowd-sourced material is:

  • anarchic
  • hard to adapt to new contexts 
  • difficult to organize
Individual visions, on the other hand, are:
  • focused
  • more easily understood and adapted
  • easy to organize in a coherent way
One of my long-term goals is to put together a Virtual Filing Cabinet 2.0, that is more complete and filled out for every unit that I teach. I've started pulling that stuff together this year, and whenever I have time (which is never) I put together a new post. I'll be posting there a lot more over the next week or so, though.