For the first time since the 2010 – 2011 school year, I will be teaching our Precalculus Honors course. Since I left the course we changed texts and five teachers in my department have arrived since then, including the colleague who I will be working with on this course. Now that it’s August, I sat down this morning with our text (we are using the Demana, Waits, Kennedy, etc) book and I have some thoughts/questions that I want to air out. I will understand my own thoughts better after writing and I anticipate some helpful wisdom coming my way through this site or over on twitter. We start off in Chapter Four of this text, doing right into trigonometry.
Things I know I don’t like
- Any formula to convert angles to arc length. Just emphasize part/whole relationships!
- Language of vertical or horizontal shrinks or stretches. I just want to talk about amplitude and period. It feels like this extra language just clutters things up.
- Inverse trig function using the odd negative 1 power. I want to write arccos x or arcsin x. Pointing out where it is and what it looks like on their calculator is a necessary nuisance.
Things I think I don’t like
- The text has an odd emphasis on the word sinusoid. I don’t know why I would want to use that word, not clear on any benefit.
- DMS notation. Why? Not sure, other than in surveying, when they will encounter this.
- Introduction of cosine as x / r and sine as y / r – I kind of want to talk about the fact that all triangles are similar and simply scale down to ‘unit right triangles’ with a hypotenuse of length 1. This feels like a natural lead into the unit circle.
Things I know I like
- An activity my colleague shared with paper plates, strings, and discovery that an arc that is equal to the radius will be subtended by the same central angle no matter the size of the plate.
- The opening day activity I am tweaking that involves the length of daylight hours as a function of days after January 1.
- Conversations I am planning on having about why it is usually sufficient to solve a triangle by knowing three facts out of six (three side lengths and three angle measures) and when it would not be sufficient.
I have taught Precalc at each of the four schools where i have worked and I always enjoy the course despite its weird, buffet style curriculum. The kids are fun to work with, sophisticated enough to have serious math conversations. We do not have the AP calendar breathing down our necks and our new schedule that includes a 90 minute class once every seven school days really lends itself to some meaningful play time in this course. I’m excited.
As always, please share any opinions/advice/questions here in the comments or over on twitter where I remain @mrdardy
Jim:
By and large, I agree with your likes and dislikes of the various elements of the Demana text and its peculiar proclivities. There is, however, one thing of real value that I think you overlook. I am referring to your concerns over the “introduction of cosine as x / r…” Although I don’t start of definitions of circular functions in quite this way, I introduce it as something similar – we define cosine to be the x-coordinate on the unit circle.
I think that it is extremely important, especially in a higher-level class like Precalculus Honors, to emphasize the notion of propositional logic and the deductive structure of mathematical knowledge right from the beginning. Kiddos often walk into Precalculus right off the boat from algebra, not really clear on this foundational way of viewing mathematics. So, when we start discussing the properties of sinusoids or trigonometric identities, we always boil our justifications down to the nature of the definition (and explicitly state that this is what we are doing).
In some sense, this is picking up students’ genuine mathematical education where Geometry leaves off.