# A Day In… Honors Precalculus with Trigonometry

## A Little Introduction for “A Day In…”

Today was the first day back from spring break. I decided it’s time to take the subtitle of my blog (better through reflection) seriously. How so? By writing a recap of one period for each of my classes, including things that went well, things that didn’t, things I can do to get better, and any other takeaways (or questions to consider) that come to mind.

This is the first installment. More will follow as time and energy allow.

## The Setting

4th Period, Tuesday, April 2, 2013

Honors Precalculus with Trigonometry

## How Things Went Down

As students walked in the room I welcomed them back from spring break. This semester I tweaked how I enter things in PowerSchool (assignments get grouped by chapter now, rather than entered individually), so I’ve been brainstorming how to help students keep track of their assignments without the same level of detail online. My most recent (and I hope final) attempt is this handout. I spent two minutes explaining how it works and how I expect them to use it. It’s pretty easy. They just copy down whatever it says under “Do @ Home” on the Daily Plan. As an example, here’s today’s:

I suppose that slide won’t make any sense unless I share the Course Outline. (I used to call it the Assignment Schedule). Students get one of these on the first day of the year. It’s my wonderfully lazy way of communicating assigned homework.

Alright, back to today. We then played SET for two or three minutes. For the first round students had 60 seconds to work quietly in small groups to find as many sets as possible in the Advanced Puzzle Mode (12 cards, exactly six sets are present, no cards are removed when a set is found). When the 60 seconds were up, each group had an opportunity to share one of the sets they found by recounting the card positions (e.g., “1-5-12” or “3-4-9”) of the. For the second round students had 60 seconds to find as many sets as possible in the Basic Classic mode (12 cards, three cards removed and replaced when a sets is found). The students did a great job, finding 17 sets in 60 seconds, a record for the week. The record for the year—held by 7th period AP Calculus AB—is 20 sets found in 60 seconds.

I was excited for today for a number of reasons. One, I typically love my job (even though I’ve had some rough stretches this school year) and I’ve missed the students after 10 days of no school (honestly). Two, I was a little bit excited about today’s lesson because I took what I thought was a rather lackluster notes handout and spiced it up tried to spice it up with some Desmos graphing action. (As it turns out, the “supplementary” handout was garbage. More on that in a moment.)

After playing SET, I grouped students in pairs, gave every student a copy of the supplementary handout, and had each pair grab a laptop (either of their own or from the laptop cart I checked out for the period). I expected this part of the lesson to last about 5 minutes, but with some typical tech-related delays many students took closer to 10 minutes to finish. (And it didn’t help that the handout lacked a clear goal. More on that later.) While waiting for the last few groups to wrap things up I invited other students to write their responses to questions 1, 2, and 3 on the board.

After the supplementary handout, we turned our attention to the notes handout. (I’ll share my frustrations with and potential fixes for the handout below.)

As we finished Example 3, the bell rang. I would have liked another 1-2 minutes to debrief, summarize, etc., but I didn’t manage class time particularly well today.

## What I Liked

It was good to see the kids again. They did a great job playing SET. Some students made important connections in spite of my poor sequence of activities. Did I mention they found a lot of sets?

## What I Didn’t Like

This could drag on for a while if I’m not careful. Time for bullets!

• Kids were bored (not everyone, but definitely some; and not just bored, but bored!!!)
• The school laptops take multiple minutes to start up (those who used their own laptops finished the entire “activity” in half the time)
• The first half of the supplementary handout didn’t have a clear purpose; the second half didn’t provide students with any engaging tasks (kids need a longer leash for healthy/rich exploration, not 60-second tasks that need teacher intervention before students can move on to the next mini-task)
• The original notes handout starts with two big ugly definitions/properties boxes (Ugh! don’t lead with this kind of thing, Michael!)
• I should point out that students had already discovered the content of the first box in a previous investigation, so I’m not totally opposed to including something like this as a summary in a later handout… But to lead with it? More ugh!
• I blew right by the second box (“These are not the properties you’re looking for…”), deciding to introduce/develop these properties with the help of a right triangle in a two-layer Cartesian/polar coordinate plane (is that even mathematically sound?), but in a very “lecture-y” manner, where my students could have developed these properties on their own with a well-written sequence of questions
• Example 1 could be improved (ideas below)
• Example 2 requires completing the square, the kids were super shaky on this, and I wasn’t prepared for their confusion (in every other year that I’ve taught Precalculus, I had taught most of the students the year before in Algebra 2 or Honors Algebra 2, so I usually have a good read on their algebra skills; I didn’t teach any of this year’s Precalculus students last year, so I’m not as in tune with their strengths and weaknesses as I have been in years past)
• Example 3 was rushed (by me, to finish before the bell rang)

## How I’ll Get Better

Alright, the point of this isn’t to stew but to reflect, and through these reflections to get better. So here goes (bullets again to avoid turning this long post into a truly gargantuan one):

• If kids were bored, that’s (usually) my fault. Here, it definitely was. I failed to give them an engaging sequence of tasks. Too often they were just waiting for the next teacher-led portion of class. With some extra time in planning the lesson, I could design something where students work primarily in groups and we only do the whole-class thing for a very brief introduction and a more detailed (student-led, teacher-facilitated) debriefing session.
• I can’t do anything about the school laptop frustrations. However, I could plan ahead and tell students when to bring their laptops to class (or have them bring them every day just in case). More and more students are bringing laptops and tablets to school. Maybe 25 to 30% now? As that grows, it becomes easier (at least for me) to incorporate web browser-based activities into class.
• The supplementary handout suffers from a lack of clear purpose. Was it supposed to help students learn some general graphing basics (toggling on/off, sliders, domain restrictions, etc.)? Or was it designed to have students explore a particular graph? Or to graph three on their own?
• When I think about it some more, the entire lesson suffered from a lack of clear purpose. Are we focusing on graphs and the impact of parameters (our conversation drifted there today)? Or is the goal to convert from polar equations to Cartesian? To be honest, I need to rewrite the entire sequence of polar lessons to give students more practice graphing, making observations about the impact of parameters in polar equations, etc., before asking them to convert polar equations to Cartesian form. So next time? Here’s what I envision: A handout with 12 images (six on the front, six on the back). Students working in pairs (each with a handout, one laptop per pair). The directions: Use sliders on my pre-made graphs to match each graph. Write down the “winning” equations. Explain how you did it. Then, on the following day we can focus on converting the equations to Cartesian form to have the “a-ha” moment(s) of “Hey, that really is a vertical line/horizontal line/circle with center (#,#) and radius #!” (Oh, reflection! How you resemble rambling!)
• Example 1 would be much better if it took over an entire page with: (1) A big fat polar coordinate plane so students can sketch what they see on Desmos or their calculators and (2) A big fat table for students to complete (theta values given, r values missing) so students can see clearly/numerically where r first becomes negative and what happens on the graph at this point. (Do I need to bring in auxiliary Cartesian graphs from the get go? Sam’s link to this applet has me wondering if that’s key early on and if I maybe wait too long to draw it in…)
• Example 2 would be much better if we first developed the properties (rather than giving students a pathetic box o’ properties (“Hey, where did those come from?” “Pay no mind, pay no mind! Back to the examples!”). This could be done with a mini-investigation.
• Example 3 would be decent if it followed the revamped Examples 1-2, provided that I move to the back of the room and have students wrestle as a whole class with how to make the conversion based on what we’ve done in Example 2.

Most of the rambling above, while helpful (I think), is focused on just a couple of lessons in a single chapter. But I’m similarly dissatisfied with a lot of the lessons in my Precalculus course. I don’t own it the way I do my Algebra 1 and Algebra 2 courses (despite their many and sometimes deep flaws). Algebra 1 and Algebra 2 are the classes for which I put the textbook on the shelf and wrote my own curriculum (here and here if you can bear to look). Did I mention the many and deep flaws?

At any rate, I know my Algebra 1 and Algebra 2 courses inside and out. I’ve wrestled with the sequence of topics, the sequence of lessons, the sequence of examples within those lessons, and I look forward to wrestling with how to turn these teacher-centered lecture-heavy courses into ones packed full of activities, investigations, explorations, rich problems, engaging tasks, etc.

Is it time to chuck the textbook for Precalculus? Can I even afford to make a move like that? I’m not exactly running around with heaps of free time these days, thanks to four beautiful little kiddos. Maybe my days of writing entire courses from scratch—flawed as they may be—are numbered? I don’t know. But I do know this. I want to get better, and I think know that I can. Reflection is time-consuming, but so worth it.

And the best wisdom I’ve heard so far on this issue (but the hardest thing for me to be satisfied with)? Baby steps. Just keep taking baby steps.

Thanks for reading! Wisdom and insight welcome in the comments as usual.

1. So appreciate the time you spent writing your post; and the depth of the thinking about your work you shared. Thank you.

2. Second the above. This is a really helpful post for me as I’m about to move into trig graphing in about a month with my 10th graders and want to bring back Desmos.

Some thoughts:

One thing that went well introducing sliders was something similar to the handout you envisioned: give the features of a particular function (in Algebra, a quadratic with certain roots) and play with the a, b, and c sliders to get those roots.

It doesn’t work yet, but if you try to convert a polar function to a table in Desmos, the error message reads “We don’t do the right thing for tables with independent theta YET” so at the rate they are adding new features, your big fat table could be in Desmos?

I lost a complete day in one class for reasons similar to those of your supplementary handout; I basically gave out a little guide to Desmos with some problems at the bottom and it wasn’t clear to my students (or me, I suppose) what the point of the 45 minutes was. Much better was a different class where each new feature was followed by a question where they were asked to use it, even a pretty straightforward one. For example: click on the graph of a parabola to highlight the points of interest and restrict the domain so that the image of the function is nonnegative.

I have kind of a silly question about kids bringing in their own laptops. I allowed it for a bit, but for a while Desmos was so new that it wouldn’t work except on the newest version of Chrome or Firefox (this was around October). Any such issues?

Anyway, great post that gave me a lot to think about, thank you.