Tag Archives: quadratic

Trigonometry with Algebra 2

Since this May may be my last month teaching math (for a while? forever? I doubt my departure from math will be permanent … more on the story behind this later), I thought I’d get into the swing of things and connect back into the theme of my blog by declaring that May is Maryland’s Math Madness Month! [To give due credit: this motto was originally thought up by my father as a way I could get kids excited about taking the Algebra HSAs, coming up now in less than two weeks.]

To kick off the month of math madness, I wish to ask other teachers of Algebra 2 with Trigonometry (A2T): how do you balance covering all of a (full-year or full-semester) Algebra 2 course in a reduced time-frame so that there is room in your course also for Trig?

A2T Collage

I suppose there are two main options for dealing with this addition of new material into an already full course: leaving out pieces of content, or teaching all the content at a faster pace.

This is something I think about every year I guess, but even moreso recently when helping students with their Algebra 2 Twilight make-up coursework and having discussions with the Precalculus teacher about what he will expect my students to bring with them to that class.

The Twilight students had question after question to answer about completing the square, which is a method I don’t cover when teaching A2T. For me, solving quadratics by factoring, “doing the opposite”, and using the quadratic formula is exhausting enough, both in the sense that it exhausts the techniques required to solve quadratics of every form, and in that my students are tired of so many methods without adding a fourth. My students have been know to complain that they are learning something new every day in my class (!).  Am I wrong for leaving out this method? Should I perhaps teach completing the square instead of the quadratic formula since it shows deeper understanding of the math involved? I don’t have enough time to do both (plus the other two methods I mentioned, which are even more fundamental).

Similarly, I treat complex numbers very lightly (last year, with the 9++ snow days, I even skipped them!). And I hear on the web about some Algebra 2 teachers teaching rational functions, which I never even conceived of as an Algebra 2 topic, since gaining an abiding understanding of polynomials is challenge enough.

So I guess that lands me primarily on the side of leaving out content. A faster-paced curriculum would leave more students lost, and I do not have selection criteria for entering the class as some teachers might. Additionally, this relates to my philosophy of math teaching, that it’s better to learn fewer things deeply than to shallowly cover everything. I try to focus my attention on the things that connect A2T to prior math and future math (e.g. “doing the opposite” as equation-solving technique and function transformations), that connect it to other subjects, that engage students with project-based learning, and that highlight big picture concepts and skills.

But it’s still a struggle, and I doubt myself (maybe I really should be teaching completing the square; maybe conic sections are more important than my Olympics research project and should replace it in my choice of topics). Especially since that Precalc teacher is counting on me to teach them certain things they will need when they arrive in that class (and just as the Calculus professor is counting on Precalc teachers to cover certain key topics).

So, A2T teachers, how do you deal with the pressure? Do you teach at hyperspeed, or what topics do you cut? [Other teachers feel free to weigh in too 🙂 ]

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Puzzling Out Some Quadratics

When teaching, I try (with mixed results) to keep my students interested in the topic at hand.  Sometimes I motivate them by giving them a project where they discover a mathematical relationship with minimal help and minimal direct instruction from me.  Or a project that demonstrates a deep connection between the real world and the mathematics we use to describe it!

Part of the Printed Puzzle

However, sometimes the topic is not all that inherently interesting, or students just need a great deal of practice to really hone a skill.  For example, the skill of solving quadratic equations.  At this point in the course, I’ve already taught them how to solve quadratic equations by factoring, by working backwards, and by the quadratic formula.  To liven things up a bit, I’ll next challenge my students to complete this quadratic equations puzzle (partly pictured at right).

The way it works:

  1. Cut up the triangles (in the picture and linked document, they are already mixed, not in their correct final locations).
  2. Give each student (or pair of students) a batch of puzzle pieces.
  3. Students match each quadratic equation edge to another edge that contains its zeros.  They can tape them together as they go.
  4. When complete, it makes a large hexagon, like this:  Completed Puzzle (small)

For extra fun, do the puzzle yourself first, then write a secret message on the backs of the pieces.  Cut it back up into its component pieces, and make double-sided copies so each tile has a letter on its back.  Only students who complete the puzzle correctly will be able to read your secret message!

For example, I have written [one letter to a puzzle piece]:  “BRINGTHISTOMRYFORACOOKIE” (Bring this to Mr. Y. for a cookie!), and then I had cookies available for the students as they finished.  When the first student finishes, brings it up and gets a cookie, the other students sit up and start working harder!

Here’s another puzzle I designed as review of Algebra1-style equations (for my engineering students, to gauge their math strengths and weaknesses in the first week of class).  It could also be used to wrap up an Algebra 1 course, or to refresh students’ memories during the first weeks of Algebra 2.

These puzzles are made with Formulator Tarsia software, which is available for free download.  Developing these puzzles is a bit labor-intensive, in that it takes a good 1.5 – 3 hours to create a good puzzle, including the secret message.

Puzzles (and food) create a nice change of pace in the classroom, every now and then.  I encourage you, gentle reader, to use either of the two I have created, and to download the software to create more yourself!  Happy puzzling!

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