Tag Archives: math

Number Systems

Binary and hexadecimal today in AP CSP:



When I was in elementary school, I was fascinated with number systems. I know I read about some different number systems used by different cultures through history in a book of number history my parents had, and was also inspired by some discussion of binary / ternary / base-4 in The Math Curse. I asked my mom to explain them, and she did.

Somehow (I forget the source, if it was a news article or an “interesting fact” in some book), I got really into the Inupiaq number system, both the symbols and the base-20 place value system for writing them. I remember writing the date in the upper right-hand corners of my papers using Inupiaq numerals each day in Mrs. McCarthy’s 4th grade class (or was it 6th grade? — she was my teacher both those years).

Always lots of fun teaching number systems! One of my favorite topics. 🙂

#day16 #apcsp #teach180 #180blog

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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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It’s Pi Day!!!.

It’s that time of year again: spring is almost here, and you can almost feel a warm mathematical breeze on the air. It’s…

Pi Day!!!.

Since the number pi (π) is approximately 3.14, and today is 3/14, today is sort of a mathematical holiday. (You may have noticed that I’ve included approximately 3.14 exclamation points above and in the post title!)

I started my celebration this morning with some coffee iced with pi-shaped ice cubes:

Pi Iced Coffee

Pi Iced Coffee

Additionally, I noticed today that I follow exactly 314 people on twitter:

Hey look - I'm following about 100pi people!

Hey look – I’m following about 100pi people!

(OK, I admit, I followed one new person today to get that to work out 🙂 )

Today, in addition to celebrating both the number pi and all sorts of mathematics, it’s time to start getting ready for the best pi day celebration of our lifetimes, which will be held in two years: 3/14/15 at time 9:26:53. This will be a much more accurate representation of pi than we celebrated just over an hour ago (at 3/14 1:59). Though perhaps we missed an even bigger party four centuries ago on 3/14/1592 6:53:58.

A few notes, links, and cool things for this pi day:

  • Math geeks can even talk about their mania for this amazing number in the form of a palindrome: “I PREFER PI”!
  • It seems that pi day is getting more popular: Companies like Oreo and GE are getting in on the action!
  • Check out this video, where a few people use actual pies to calculate pi (from http://www.numberphile.com/videos/pie_with_pies.html) :

Lastly, I hope you all can read (and would agree with) the following t-shirt, tweeted by NSBE:

Picture from NSBE

Picture from NSBE

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Happy 12-12-12 Day!

Perhaps it could be called duodecimal day if you look at the numbers, or ternary day if you look at the digits. In either case, have a wonderful 12-12-12 day!

After being treated to a dozen similar days over the last few years (here, e.g.), we must now alas face a desert stretch of a few weeks longer than eighty-eight years until the next time the month, date, and year will all align.

Though at least we’ll have 11/12/13 next year, and 12/13/14 the year after. And of course the best pi day of our lives on 3/14/15 (at 9:26:53am). So I guess, even without repeated numbers, we still have a few good years ahead of us 🙂 .

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Pi Day Arrives Again!

Once again, as the Earth travels its circuitous path around the Sun, we arrive on that day we call pi day.

Just two weeks ago we had Leap Day, to help correct for the fact that our spheroid’s spins and orbits don’t match up evenly with one another. Which means that, on this particular 3/14, our position along that route of revolution most closely matches up with that of pi day a quadrimular period ago–3/14/2008–or that now past by two quadrimular time-spans–3/14/2004. Pi Day 2004, incidentally, was the time my emails took a dramatic turn (I had sent a pretty bland pi day message to friends and acquaintances in 2003); if you’d like to revisit that missive and recall a few handy mnemonics for remembering pi’s digits, click here.

This year, I’ve tried to make pi day last all week by highlighting and celebrating math, its importance, and beauty, more consciously than I usually do in the day-to-day of my life and math classes. We’ve talked in class about phi and the amazing fact that Fibonacci numbers and patterns can be found in plants. Today, in honor of pi day itself, I did the following:

  • Wore a pi day temporary tattoo on my forehead (making me look a bit like my profile picture on my About page, though with the pi’s colors inverted)
  • Wore this pi day t-shirt [ a few other cool pi products, including a pizza pi cutter (how awesome is that!?), can be found here]
  • Brought in pie for my students – my current students ate some during class, but I also invited them and former students to come back around pi moment (3/14 1:59pm) for another slice
  • Listened to this song
  • Watched this video
  • Had students work on a pi day online scavenger hunt (this one’s been a tradition for several years)

Wearing the t-shirt and tattoo were a good plan, to get not only my students but other random students in the hall talking about pi today! I had students come up and ask me what was that thing on my forehead (a chance to explain/recall the number pi), others who recognized it but asked why pi was on my forehead (a chance to explain the glory that is pi day), and some who upon seeing me said, “Oh, it’s pi day, right!”. It was a conversation starter, even if I looked like a crazy math fanatic!

Hope your pi day was similarly eventful and pi/e-ful too!

I leave you with two awesome things:

Photo © Shannon at a periodic table (used with permission) - click the photo above to go to the original photo and blog

A pi apple pie, by blogger a periodic table. I wish I had the talent to do that; this is the most amazing pi pie I’ve ever seen!

A video by mathemusician Vi Hart, celebrating both pi day today and tomorrow’s Ides of March by speaking in Iambic pentameter about the possibility (probability?) of finding the works of Shakespeare encoded within pi’s digits – very cool!

Though but an hour remains here on the east coast (though, after all, why not take the whole week to celebrate?), here’s hoping the rest of your pi day is mathematically delicious!



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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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