January 21, 2013

PreCalc: Semester 1

As midterms are this week, we have reached the halfway point of the school year.  Hard to believe it!  This seems like an ideal time to look back over first semester.  In my school we start PreCalculus with trigonometry.  Students only see basic triangle trig in Geometry so this is a new topic for everyone.  In the past teachers found themselves sucked into review and losing the strongest students at the very beginning of the year, they decided starting with new materials is the best way to avoid this.  To be sure all students were starting on equal footing when it comes to the basics of graphing and manipulating graphs we had the honors students complete a summer assignment.  Then, the students who struggled came to me outside of class for extra help, and the ones who had already mastered their families of functions moved straight into Trig.

The very first day of class I had students make up a random survey question and survey their classmates (getting to know you activity with ulterior motive? check).  I then distributed a large circle with a point at the center to each student and instructed them to create an accurate circle graph of their data (I none too subtly pointed out where various supplies are located in the classroom).  Finally, I asked students to precisely find the arc length of each arc in their circle (in centimeters).  Students played right into my plan as if they knew exactly what we were doing and asked about using degrees to measure arcs like they did in geometry.  Enter the radian!  This lesson would have been perfect if I'd had radian protractors, but we did well enough without them (plus, they didn't even exist in September).

From here we continued into your basic triangle trig, build the unit circle, graph trig function sequence. Nothing particularly spectacular to say here.  I wish I'd had Fouss' awesome unit circle at the time, it's much more organized than the ones I was using.  After we had finished our study of the unit circle, one of my students had the suggestion that if they could fill in an entire unit circle they should get to use one on the tests (I'd been giving them blank ones to use, but his complaint was it took too long to fill in).  This was a totally valid request and so I gave everyone this option for the midterm, next year I'll do it earlier.

When we got to identities, students struggled.  For valid reasons I outlined before.  But also because I forgot to have them ever use what they struggled to prove!  Most of the time, we prove something so that we can use it, not just because proofs are fun.  I realized my mistake when I was glancing over the midterm some classes used last year.  Next year we will alternating proving identities and using them to solve problems.  This will quiet some of the pleas for numbers (my students really missed arithmetic during this unit) and will give them a sense of purpose in amongst all of these challenging, many step problems.  For this exact reason, laws of sines and cosines were received with cheers- we actually used something we proved and there were numbers again!

Our final unit of first semester was inverse functions.  This was part review, part extension.  Most students did well and the biggest issue I encountered was students trying to solve everything in their heads.  I'm hoping that second semester will be similar since we will now move into topics that should be familiar but we will build up and out and around their current knowledge.

I used a version of Standards Based Grading this year where students assessed on each topic twice.  We do short quizzes on one standard and tests on a few standards at a time.  I'm generally happy with how that has been going.  The honors students are awesome about taking advantage of the retakes and they appreciate only having to retake one section of a test (as do I- less grading!).  We do a variety of in class activities for each unit (called tasks below).  Some activities are longer and require out of class time, those are collected for a grade.

Quarter 1:

  • Task: Summer Assignment on Families of Functions
  • Radians (Task: circle graph)
  • Evaluate Trig (Triangle Trig, Unit Circle) 
  • Graphing Sin/Cos (Tasks: ferris wheel, writing prompt) (better method for describing shifts and stretches - use next year)
  • Writing Trig Equations (from graph or description)
  • Graphing Trig Functions (using graphing calc) (Task: make a picture using trig functions)
  • Evaluating Inverse Trig Functions

Quarter 2:

  • Proving Identities (Task: geometric and algebraic proofs)
    (skipped applying them until the midterm review)
  • Law of sines/cosines (Task: derivation of each law and area formulas)
  • Law of sines/cosines Applications (Task: writing prompt)
  • Triangle Area (forgot Heron's formula until after test, should be part of applications)
  • Find/verify inverses graphically (includes restricting domain and identifying one-to-one functions) (Task: use tables, equations and graphs to determine characteristics of identities)
  • Find/verify inverses algebraically

I thought I'd done a terrible job with blogging about PreCalc but it wasn't actually so bad.  I'll try to fill in the holes of how I implemented geometric proof (a post I started ages ago but never finished) and the inverse function unit.


  1. Neat! I find it sort of fascinating that you were able to spend several months on trigonometry. In our Pre-Calc curriculum, it's just one of four strands (Trig, Exponent/Logs, Poly/Rationals, and Function Mashup). In the Grade 11, I've got less than a month to hit Trig expectations (granted, in 75 minute periods, with, it seems, fewer overall items).

    Also interesting that you go in reverse to me (perhaps because radian measure isn't until Grade 12). My first go through, when I brought in the unit circle first, I lost some students that never really come back for the triangles. So the last couple times, I've done triangles first - including obtuse triangle cases - then generated the unit circle to attempt to explain the ambiguous issue with inverse sine, and on from there. Still not sure if it works, I always feel like the chicken and the egg. (The circle explains the angles, but the angles are what generate the circle...)

    Anyway. Yeah, an "identity crisis" isn't unusual either way, it would seem. All the best for the rest of your year!

    1. I guess we spend so long on trig because it only gets a brief introduction in geometry and then students go from here to AP Calculus so they're on a time crunch. I actually have no idea if we're pacing this well to get through the list we made at the beginning of the year, which looks similar to yours but since the other Honors PreCalc teacher is also the AP Calc teacher, I suppose whatever happens will be fine for this year. Then at the end we can reassess, which of course is my main purpose in writing these reflections. It's always a pleasant bonus when someone else finds them interesting!