Tag Archives: geometry

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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Math is in the Air!

Math is in the air this week as we close in on that most special of mathematical holidays, pi day.

Pi Day is only a week away, and I for one can’t wait! I’ve emailed our school secretary so pi day will be included in our weekly bulletin. I’m trying to consciously make students appreciate some of the cool things about math (I try to do this all the time, but sometimes get stuck on autopilot teaching procedures and projects).

In preparation for Pi Day, I’ll be posting some links to amazing mathematics over the next week.

To begin, you all remember the interdisciplinary geometry-art project on fractals I did last year? Well, these middle schoolers have gone even deeper into the world of fractals and produced some beautiful works of art, as part of their Fractal Club! Go to this link and check out the video. (Thanks to Ceilon Aspensen for sharing the link on facebook.)

Have a great pi week, and I’ll see you again soon!

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November in Geometry

Although this is my sixth year teaching, I’ve been struggling with classroom management issues this fall in my last-period Geometry class. So we haven’t been able to do some of the cool projects I talked about last year (click the Geometry tag to see more). And a few topics we haven’t been able to delve into at quite the same level as I could with a more-motivated group of students.

A few details on some topics we’ve worked on during the past month or so:

The lesson on three-dimensional polyhedra went fairly well for the first two parts. Students constructed polyhedra from nets and by building their skeletons out of gumdrops (vertices) and toothpicks (edges). They discovered the relationship between vertices, edges, and faces found by Euler (V+F=E+2). But when I tried to bring the whole class to proving that only five regular polyhedra exist, I lost 80% of the class. I don’t know if it was too many steps, too long for their attention spans, an aversion to the logic of proofs, or the overall class dynamic. I don’t believe the math was too complicated for them (it just has to do with angles in regular polygons, spatial relationships, and our previous topic – tilings of the plane). I provided a sheet for taking guided notes. But much of the class turned that sheet in without having taken any notes.

Some of the more successful lessons have been a few that tied into what my students were learning in their engineering class. In late October – early November, my sophomore Geometry students were building and analyzing truss bridges in their Principles of Engineering course. Several teachers got together to plan lessons in various subjects that tie into the topic of bridges. In October, near the beginning of that unit, I did a lesson on the strength of various shapes. Students tried to use paper to hold the most books at least one inch off the table. They constructed a bridge that could hold the most rolls of pennies, using just one index card. And another bridge of multiple index cards, designed for length.

A couple weeks later, after they had developed designs in the engineering class, I had them analyze some of the geometry of triangles. This connected their bridges to what we were currently talking about in Geometry, with triangle congruence, proof, naming, and the Pythagorean Theorem.

Bridge Geometryhttp://www.scribd.com/embeds/73632593/content?start_page=1&view_mode=list&access_key=key-1xgwxwz3mcqb0iig9brt//

We’ve also learned about isosceles triangles, angle relationships, and circles in the past month.

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Reminders: Please support my moustache & Baltimore students by donating, and please support my partners in Baltimore’s NaBloPoMo by visiting and commenting:

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Day Before

Attendance was light today, as always on the day before a holiday like Thanksgiving or Christmas break. My eleventh grade engineering class was close to being there in full force [of those who are normally there], which I was proud of. That group I have developed a positive relationship with, and made clear I was expecting them today, and that we would be learning/doing new things today. My tenth grade geometry class was much fewer in number. They are (relatively) new to the program/academy, and I haven’t built up as good a relationship with them yet. It is also last period – some who had come earlier in the day may have left out by then for an early Thanksgiving break.

A few students who came today said they weren’t doing anything in other classes, why am I making them do work? I wish that, like other school systems, we had today off. Or at least a half day, like other systems. But if we are here and have school, it is a normal day of lessons: we are learning new things, and the work we are doing is important. I try to emphasize that to students in my explanations, as well as show by example.

In Geometry, we completed an exploration into how to construct a circle that goes through three predetermined points. This is one of the top three skills related to circles that we learn. Those who finished early played a game of polygon capture. In the small setting, I was able to really push their thinking about why certain things were true (i.e. proofs of what/how we were constructing circles).

In CIM, we reviewed reading programs that control a robotic arm, worked on analyzing them critically, both answering questions and filling in missing parts of a program. This is a skill vital to their understanding of the work we do in the class (reading, writing, and analyzing programs of different types is probably 75% of the course material) and therefore also important to the final exam that can help them earn college credit for their engineering coursework while still in high school. After that, students added to their online portfolios.

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Reminders: Please support my moustache & Baltimore students by donating, and please support my partners in Baltimore’s NaBloPoMo by visiting and commenting:

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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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The Pythagorean Theorem

Possibly the most famous theorem in all of mathematics. Over 300 distinct proofs of this theorem exist, including one discovered in 1876 by future president James Garfield. [Unfortunately, his mathematical prowess did not protect him from the assassin’s bullet.]

I introduced the Pythagorean Theorem today in class. Here’s how I’ve taught it before:

  1. State the theorem. For every right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs (a.k.a. for a right triangle, c^2 = a^2 + b^2). Almost every student has seen it before, either in middle school, in algebra, or in engineering.
  2. Do some practice questions involving solving for the unknown hypotenuse.
  3. Do some practice questions involving solving for an unknown leg.
  4. Do some practice questions involving Pythagorean whole number triples, where you have to figure out the third number without knowing whether it’s a leg or hypotenuse.
  5. Introduce the Pythagorean Theorem converse and do some practice questions of figuring out whether a triangle is acute, right, obtuse, or nonexistent.
  6. Have students research a proof of the Pythagorean Theorem, come to an understanding of it by reading it and working out the pictures and/or algebra, then write up a mini-report. Here‘s the project description.

Anything vital (or just extremely cool) that I’m missing? How do you teach the Pythagorean Theorem?

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