2 is better than 1 (sometimes)
I’m a fan of visual context over simple memorization of formulas. Another pet peeve of mine is the requirement that denominators be rationalized without reason. A good conversation on rationalizing the denominator takes place over at “Why am I teaching this?”. One place I’ve see rationalized denominators lose their context is in a unit circle. I get that it’s a standard. I get that the angle radian measure makes most sense with a radius equal to one.
A friend of mine, Jen Silverman, makes some great protractors that help with this radian angle measure as well. I think my frustration comes from the idea that the unit circle is a beautiful overlay that simplifies so many interactive relationships between the trigonometry ratios. Too often students never come to this realization and instead resort to tricks.
Here’s another post using the same trick.
I used to try to get students to think of the Unit Circle key segments as just a small set of lengths that interact in different ways.
Some applets to help in relating the unit circle to the cartesian form of basic trigonometric graphs:
While others simply go back to memorizing the rationalized denominator forms.
Still, I feel the abstraction away from Trigonometric Ratios loses the relationships within the right triangles created by these points along the unit circle.
So my proposal is: Introduce Circle Trigonometry with a radius of 2. This would double all the segment lengths. Trigonometry ratios would relate more directly, and students could redraw and label the pieces more intuitively [it’s pretty hard to imagine what Sqrt(2)/2 looks like compared to Sqrt(3)/3)]. I haven’t built the applet for this, yet. Expect it to show up in a future revision of this post soon. If you’ve ventured this way before, or you’re interested in encouraging/discouraging me from doing so, please comment below.