This has been a busy week in terms of placement. Between the new teacher orientation and trying to get a cohesive lesson plan for the first three weeks of school, there’s not much time for my college classes. Luckily, my classes are not as time consuming as they are mentally challenging. So other than my brain being fried I’m doing pretty good.
I’m co-teaching precalc this year. I think it’s a good thing, as it is the class I TAed during my undergraduate career. Of course, teaching it in high school is very different than teaching it in college, but I feel like I understand the concepts better than I would if I hadn’t looked at the materials in six years.
One thing that seems to be a main focus of this class is the emphasis on identifying and describing graphs. To do this, the class uses an acronym called Dr. Id Zimmah:
Domain
RangeIncreasing
DecreasingZeros
Intercept
Maximums
Minimums
Asymptotes
Holes
I am always weary of acronyms, as I feel they can mask the significance of the concepts. I believe the major problem with precalculus is that it really only exists to support calculus. As a result, it seems like a lot of memorization without much understanding of its use until you reach higher levels of mathematics. between Dr. Id Zimmah, function families, laws of exponents and trigonometric identities and everything else the students have to learn, I think precalc does a fine job at making students never want to take calculus. I suppose this will be a challenge throughout the year.
To introduce the students to functions, I think I might show them this graph:
I could explain to the students why I liked or didn’t like math. I might ask them to find when I liked math classes the least (minimums) or the most (maximums) and come up with reasons why that might be. I could point out that shortly after 11th grade my enjoyment spiked exponentially. I could then explain why that was and tell them how my goal for the class is to figure out a way to shift that graph to the left a few units (horizontal shift) so they can see why math is so awesome without hating it first.
Cool, I’m student teaching Math Analysis which has elements of pre-calc. IMO, acronyms do not necessarily preclude deep understanding (e.g. SOHCAHTOA). They simply help with recall. I do like your graph; mine would have both relative & absolute maxima and minima though.
Boy do I feel ya. I taught precalc this summer and will again this fall and struggled to find some way of making it an end in itself. I didn’t succeed, but got a little closer by simply pretending it was a calculus course and every time we got stuck because we didn’t have the precalc support, we went back and learned what we needed to move on. It wasn’t a perfect solution, but those students are as prepared for their calculus class as I can make them. Good luck!
Interesting solution. I’m curious as to your general outline for the summer and how this would look in action. Perhaps it is my inexperience, but I feel like some concepts (like function families and laws of exponents/logarithms) are not directly used in calculus but show up implicitly, so it would be hard for me to come to a moment where I’d say “lets go back and learn this.”
I am lucky in a sense as my class does not use a text book, so we are free to tailor the class in any way we want. We have changed the materials a lot from last year and so far I feel like things are going pretty well. Then again, it’s only been 6 days – four of which were spent making sure everyone was at a basic level of graphing linear functions, solving equations, and simplifying expressions.
I hope your class is going well still. Certain things come from calculus problems more naturally than others. Typically, I use a calculus problem to hook the students into talking about, say, logarithms. When computing (many) simple examples, I try to coax the students into noticing things like the exponent/log laws. Then I’ll take it back to the calculus and show them a problem where those laws are helpful (e.g., logarithmic differentiation). I don’t expect the students to follow everything I’m doing on the calculus side, I just want them to be exposed to an example where the laws they’re learning are demonstrably useful. Of course, not everything goes back to calculus. Log laws are inherently interesting for other, more elementary, reasons and these are the examples I stress that the students understand.
Anyhow, I’m enjoying your blog quite a bit and I hope you keep putting up such quality material for us. It is a pleasure to read.