Tag Archives: algebra

Roots of Unity Final Projects


These are the results created last November for the art/math integration project described here.


By the way, happy pi week everyone!

Pi Week

Pi Week

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Art Integration: Roots of Unity & String Art

I talked with an art teacher this afternoon about ways to integrate art and math into a project. She had some great ideas, plus we came up with more ideas in the course of our discussion, many of which I plan to try for Algebra 2 or Precalculus (both which I teach this year, fall and spring respectively). Geeking out while discussing the intersection of math and art reminded me of this awesome collaboration and its result from a few years ago!

Our first idea (in terms of implementing soon) was some colorful string art crossed with a discussion of the roots of unity, since my students are (today) using and graphing complex numbers for the first time. Math teachers, art teachers, and any interested others, check out this rough draft of the project and let me know any thoughts and advice:

Roots of Unity Project


Colorful "string" art on the 17 th roots of unity

Colorful “string” art on the 17 th roots of unity


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Phi Project

So, as I explained yesterday, I decided to create a project centered around golden ratio, phi (φ) ≈ 1.6180339887, and the associated Fibonacci sequence 0,1,1,2,3,5,8,13,…. My classroom has computers, so I had students go back and forth between watching parts of Vi Hart’s videos (123) on the subject, and doing or reflecting on something mathematical, artistic, or biological.

Here is my project:

I think it’s a little bit lighter weight than some of my other projects. But it does connect to our work with quadratic equations (which we are just wrapping up). And it gives my students a chance not just to see math in the world, but also to think about why our world is mathematical.

The students seem to enjoy working on the project so far. Either that or they just liked the pineapple we ate (after, of course, counting the spirals on it!).

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Golden Ratio

I can’t believe I’ve taught math for almost six years and not done much with the golden ratio, phi (φ) ≈ 1.6180339887, and the associated Fibonacci sequence 0,1,1,2,3,5,8,13,… (add the previous two numbers to get the next number, so 8+13=21 c0mes next)! Especially when so much of my own research in college was connected to phi. You see that spiral up in the blog heading? It’s related to the more famous golden spiral and Fibonacci spiral,

Fibonacci Spiral

but it’s actually something never-before-seen I discovered about six years ago: the Yates Golden Diophantine Spiral. It has the special property that, when centered in a coordinate plane at (φ,0), it has x-intercepts at precisely ratios of consecutive Fibonacci numbers! For example, at 3/2, then at 5/3, then at 8/5, etc. That’s something those other spirals can’t claim!

And then there’s my research into continued fraction representations of irrational numbers, of which phi is the simplest:

It turns out that every quadratic irrational –like φ = (1+√5)/2– has a continued fraction that repeats periodically at some point (e.g. √7 = [2;1,1,4,1,1,4,1,1,4,…] has period length of three because 1,1,4 repeats). Now this by itself is pretty cool since quadratic irrationals’ decimal expansions continue forever but never repeat! Anyway, seven years ago during the summer, a group of three other undergraduate students, myself, and our faculty advisor were able to find a way to write alternative continued fractions for every quadratic irrational number with just a single number repeating every time (period length one). This is true even when the standard continued fraction has a period a million numbers long, which, when you think of it, is pretty surprising! Our results were published in the Journal of Number Theory. In fact, our first step toward discovering these results was experimenting with some Fibonacci identities like these.

So, it seems like a natural topic for me to include phi and Fibonacci in my classes, especially with all the geometry involved in golden rectangles and spirals, and the quadratic equation that generates the algebraic number phi. But, aside from a short detour my first year teaching Algebra I, where we were talking about patterns including the Fibonacci sequence, and my students asked me to explain my research, I have not done anything with phi in teaching Geometry or Algebra 2.

That’s all about to change. I was inspired by the following video (click through to see parts 2 and 3 as well):

I’ve developed a project centered around that video, that connects Algebra 2 to Biology and Art. More on that tomorrow. 🙂


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SBG Update

OK. I haven’t talked much about Standards-Based Grading (SBG) since last September. I’ve changed some things about how I implement SBG, which I shall describe here, along with how things went.

Geometry, Fall 2010

Here’s how I set up my Geometry class procedures–SBG-related, anyway–for my fall semester class. We had thirty-two skills, which were mostly assessed in class three times each. I converted students’ SBG grades into a percentage for reporting out by first averaging them and then using a cubic relationship (the third one from this post) to match that average to a traditional percent scale.

The students pretty much did not initiate any re-assessment. Less than five skills were retaken outside of class time. I think for the most part they were confident that, between the next two in-class retakes, they would improve just by being in class and doing the daily assignments and warm-up reviews. But even if they didn’t improve (or even regressed), the students did not seek out help nor did they come in to re-assess. They were content just to let the 1 or 2 be averaged in. And mostly their scores were high, in the 80s [those who attended class anyway].

Algebra 2 with Trigonometry, Spring 2011

For several reasons, including combating this apathy about un-mastered skills, for my spring Algebra 2 with Trigonometry (A2T) class I made a major change in how I converted SBG skills grades into percentages. Instead of averaging grades and then scaling, I set cut-offs of how many skills need to be mastered to earn a particular grade. Rubric here. This was also in part prompted by criticism of averages as being antithetical to the nature of SBG by the Science Goddess and @mthman.

Other than this big change in grading, I mostly kept the same procedures from fall to spring. There were more skills (blog posted here in draft form, final form here), so we only visited each skill twice in class via our weekly quizzes.

Many more students re-assessed this semester. But many waited until it was too late to seriously improve their grade. For this reason, and because we only assessed each skill in class twice instead of three times, grades were generally lower this semester than in my fall geometry class.

And there were still a few students who seemed lackadaisical about their lack of skills mastery and never came in to re-assess even though they were failing or scraping by with a D-. Perhaps I didn’t motivate them well enough by extolling the glories of SBG and how it revolutionizes grading? Can anyone refer me to a motivational-type speech you use with your students about how SBG helps them?

Another new piece is that I had set some prerequisite skills (mostly from Algebra 1 but a few from Geometry) that in my mind are so fundamental and which the main skills of A2T build upon. To promote my students’ review of these prerequisites so that I could help them build new skills upon that foundation, I required that students master these prerequisites before they would score any points for core skills (again see the rubric I used). But many students did not really take the prerequisites seriously. By which I mean, they did not review work from Algebra 1, and make sure all prerequisites were mastered within the first week. Some waited until the middle of the quarter to remediate the prerequisites, others until the last week of the quarter–finally being motivated to fix them then so they didn’t fail.

Is it fair for me to set this separate category of skills without which they cannot earn any points? I thought so, since they should not be in an Algebra 2 class without being able to understand key Algebra 1 material. If it is fair, how can I make the students understand the value of the prerequisites in supporting further learning and make sure to really review and master them the first week? Could I refuse to let them sit for other quizzes (on core A2T skills) until their prerequisites are mastered? Or should I get rid of the distinction between core and prerequisite skills entirely?

I welcome your thoughts and criticisms.

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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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Today is our third snow day in a row. While I am impatient to get the new semester started, I understand that many people are still without power (according to BGE, this storm while only 10″ created more power outages than our 45″ last February), the streets and sidewalks are very slippery, and some side streets are still not plowed. Today would also have been not so good as a kick-off to the new classes and new semester, with the combination of snow and Friday-after-four-days-off creating low attendance. So I look forward to meeting my new students and having an amazing first day on Monday.

In my time off, I’ve created multimedia syllabi with Prezi for both my Algebra 2 / Trigonometry (A2T) and Principles of Engineering (POE) classes. Also, I appreciate the feedback and comments on my A2T and POE skills lists, and I’m still making modifications to those lists, so any additional conversation is welcome!

Have a joyful day!


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