A Good Day in Precalculus

After struggling a bit earlier in the week, it was nice to end with a positive experience  in Precalculus on Friday. There’s nothing profound in the sentences below, but I still think it’s worth writing down, if for no other reason than increasing the likelihood that I recall some of this as I plan future lessons.

Context (for the Year)

We made the move to block schedule this year. After years of being opposed to the idea, I recently softened my stance and even began to think a block schedule approach might be helpful. I’m still learning the ropes, but the “early returns” in my own classroom (from me as well as my students) are positive.

Context (for the week)

We’ve been working on trigonometric properties and identities. I had a rather disappointing experience in Precalculus on Wednesday, and though I can’t recall the details of last Monday, I’m pretty sure it was far from amazing.

The Daily Plan

As students shuffle into class each day, I throw “The Daily Plan” on the projector. Friday’s looked like this:


Mini Exploration

Some of what I disliked about my lessons from earlier in the week related to the all-eyes-on-me approach I took. So I decided to start Friday with a miniature exploration. Again, nothing profound, but something that would:

  • Provide students with a brief review of the properties they would need to have at-the-ready for today
  • Give each group (or pair) of students an opportunity to proceed as quickly/slowly as they needed to
  • Challenge students to express their thinking in writing and in discussion
  • Allow me to wander through the room, checking progress, lingering with students who needed extra support (which I tried to supply via questions, not statements)
  • Require students to do some individual and small group thinking and wrestling before our whole class discussion/recap

Here’s what the handout looked like:

page-one page-two

And here’s a link to the two-page PDF (in all of its non-glory).

Reordering Task

After debriefing the mini exploration (which included a whole-class conversation and a “puppet volunteers” work-through of Problem 6), we moved on to a reordering task. It’s an idea I had been thinking about for a few months (years?), but one I had never put into action. After using this on Friday, I think I’m sold on its quality—at least for some problem types.

Here’s a look at the student page:


And a link to the handout, for closer inspection.

Properties Quiz

I should have done this each day for the last three days, but my moments of genius are few and far between. On several occasions in class we’ve described the properties as “puzzle pieces” and proving identities as “solving a puzzle.” If you don’t have the basic properties memorized, it’s like trying to do a puzzle with pieces missing. Apparently that metaphor was insufficiently inspiring, as many of my students spent little to no additional time at home committing the reciprocal, quotient, and Pythagorean properties to memory. And this lack of recall presents a problem for the work at hand.

An older version of me would have tried to remedy that problem by lecturing the class about the importance of blah blah blah. On Friday, I decided to skip over that part and instead gave students a few minutes of class time to work on committing these bad boys to memory. “5/5/5” on the daily agenda meant five minutes of silent and individual study, five minutes of partners quizzing one another verbally and/or in writing, and five minutes of do-the-best-you-can-on-your-own quizzing. I may have only given them 3 or 4 minutes for each stage, but the results were great. Most students now have the “puzzle pieces” in hand, and those that do not know exactly where their weaknesses lie.

Visual Patterns

In a block schedule setting, I’m finding that focusing on trig properties and identities for the entire class period is just too much of the same thing. To mix things up—and to plant some seeds for an upcoming functions- and graphing-heavy chapter—we worked through a Visual Pattern (our first one in months). As an aside, I wish I did a better job of sticking with my start-of-the-year resolutions (e.g., “Visual Patterns will be a regular feature in such-and-such class this year.”) I suppose it’s not too late to bring it back into the mix…

Personal Takeaway

If my second sentence in this post is going to be true, I need to nail down why I think Friday was better than the other days in Precalculus last week. I think it boils down to two things:

  • In designing the lesson, I endeavored to make my students the key do-ers throughout the lesson (whether via thinking, writing, arguing, sorting, explaining, or defending)
  • In an effort to maintain student focus and fight off the feeling of the class “dragging on and on,” I provided students with several distinct (though still related) tasks

As I create my next set of daily plans, I’ll try to keep these little victories from Friday in mind.

Comments 3

  1. When you say “we worked through a visual pattern” what do you mean? Last month in one of my classes we focused on visual patterns at the beginning of class each day but I’m wondering what that looked like in your classroom.

  2. Post

    Laura, I glossed over that since it would probably have doubled the length of the blog post to include all of the details of our approach. The short version: I project the image, ask students to draw the next step, then to sketch out the 27th step, then build a table of numbers for steps 1-5, 10, 27, and n. Try to make connections between the visual, numerical, and algebraic structures. Then use our expression to write and solve an equation to answer a “what step will have such and such number of items” style of question.

    I may write a post about how Visual Patterns plays out in my classroom sometime soon to solicit feedback. My general frustration is that I’m still stuck in an all-eyes-on-the-same-page approach. Having trouble breaking out of that…

    In the meantime, would you be willing to share how you use Visual Patterns in your class?

  3. I actually used exactly the same approach as you did. At first my students were scared or intimidated just to draw the next step. Eventually we moved past that and they even became comfortable filling out the table for n=1, n=2, n=3, etc. But when I asked them to write an expression for n they really struggled! They were stuck in trying to find a pattern of how to get FROM step 2 to 3 or step 3 to 4. Theh would say things like, you just have to add 5 every time. They had a very hard time understanding that the goal was to create an expression for any number n. Any suggestions on that part?

Leave a Reply