Thursday, May 26, 2016


In the Nature of Mathematics course we were talking about China today. The main activity was the students trying to figure out the nets that make the Liu Hui solids, which I learned about from Jennifer Silverman. It's wonderful seeing the students engage in 2D/3D thinking. Today we started with the Tangram instead of Magic Squares, because the students had been frustrated with Archimedes Stomachion. They were challenged, but successful, and we got onto some neat puzzles in some groups using multiple sets and making squares of different sizes.

Here's the handout:

But what I wanted to write a note about was the idea of commentary. Mathematics in China followed a bit different path than in other ancient cultures, perhaps because there was more prevalent instruction. Lost is the origin of their ancient text, Nine Chapters on the Mathematical Arts. Instead of the advance coming from a collater, the big jump was Liu Hui writing a commentary on the text. To get the feeling of it, I asked students to solve one or more of the sample problems:

Chapter 6:12. A good runner can go 100 paces while a poor runner covers 60 paces. The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit. How many paces does it take the good runner before he catches up the poor runner.

Chapter 7:1. Certain items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins, and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items.

7:18. There are two piles, one containing 9 gold coins and the other 11 silver coins. The two piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins. Find the weight of a silver coin and of a gold coin.

Chapter 8: 1. Top-grade ears of rice, one bundle, medium grade ears of rice, two bundles, low grade ears of rice, one bundle, makes 39 dou. Top-grade ears of rice two bundles, medium grade ears or rice three bundles, low grade ears of rice one bundle makes 34 dou. Top-grade ears if rice one bundle, medium grade ears of rice two bundles, low grade ears of rice three bundles makes 26 dou.
And then to write a commentary on it. This is new territory. So I had them share in groups, and pick someone to share with the class.

Mostly what they shared was there a solution, so I asked a commentary type prompt after each. Usually I'd be hesitant to put people on the spot, but these are senior students and we've had a pretty open classroom culture so far. I asked about solving with different representations, how you would describe in general the solution method, and an extension question about the mathematics. As they got into those discussions, it occurred to me that this might be a good framework for thinking about writing in our foundations classes. As the students discussed the idea of commentary, they noted that it seemed like a good way to draw attention to the idea of generalization, and a support for student reflection.

So thanks, Liu Hui! We'll see if I can get students writing their own commentaries on the mathematical arts.

Wednesday, May 25, 2016

Polygonal Spiral

I have a fascination with spirals. Exhibit 1, my GeoGebraTube materials, searched for spirals:

That's some of them...

By the way, if you haven't been following Megan Schmidt's spiral adventures, you're missing out.

I've been interested in polygonal spirals for a while, but then my student Andrew's tessellation got me thinking again.
It's Archimedean since the spirals have a kind of constant width. At first I thought it was triangular, but it's clearly hexagonal. Interesting that two of them fit together to fill the space... that's something I need to think about more.

To build them in GeoGebra I made a list of N directional unit vectors, and then a scaling sum to get a spiral of points, like 1*v_1, 1*v_1+2*v_2, 1*v_1+2*v_2+3*v_3, ... with a modular function to reuse vectors in order.

Then I connected up to points 1/Nth of the way to corresponding vertices to make trapezoids. Then I rotated them N times, around a center I located by intersecting the perpendicular bisectors of points I wanted to correspond.

GeoGebra geek paragraph: The colors are the hardest thing to get, because in GeoGebra you can't set the color of different elements of a list. My current workaround was suggest by someone on the GeoGebra forums a few years ago, and I keep reworking it.
Execute[Sequence["Delete[R_{"+i+"}]", i, 1, oldN]]
Execute[Sequence["Delete[R_{"+i+"}]", i, 1, oldN]]
Execute[Sequence["R_{"+i+"}=Element[list8, "+i+"]", i, 1, N]]
Execute[Sequence["SetDynamicColor[R_{"+i+"},  "+i/8+","+(.5+(i*(-1)^i)/(2*N))+","+1-i/8+",.75]", i, 1, N]]
Essentially you constantly create and destroy objects from the elements in your list, and then set their color.

Usually I have a list of color names and run through them with the SetColor, but in this one I wanted a higher opacity, so made up a way to set R, G and B values in the SetDynamicColor command. 

Here's the result! It was hard to think about how else to dynamicize it, since it's a pretty rigid structure. Any ideas?

On GeoGebraTube, too.

Tuesday, May 24, 2016

Khayyam's Cubic

So I'm going to try to think aloud about something I asked my students to do today. It's on this activity, about one of our greatest human minds (IMHO), Omar Khayyam. No reason for this guy to be less famous than Leonardo.

How would you try to solve \( x^3+x=4 \)?

So first I think about some concrete values. 0+0=0, 1+1=2, 8+2=10. OK, one solution between 1 & 2. No negative solution. Accessing calculus (which seems like cheating) derivative is \( 3x^2+1 \) so monotone increasing. I think that means this is a root with multiplicity 3, as one root, two imaginary is an 'S-curve' as my students say with the one root. (I should look into that at some point.)

Algebraically, I think about factoring as is, which seems like no help. If I'm right about \( (x-a)^3 \) for some \( a \)... that means \( x^3 - 3 a x^2+3 a^2 x-a^3=x^3+x-4 \). So is \( a=4^{1/3} \)? That doesn't work! And then there's no way for \( 3 a x^2 \) to be zero. So I take back what I said about multiplicity! (I really do have to look at that more.)

So next I would look at numerically grinding it out. Something closer to 1 than 2 and proceed from there.

To solve it with a graphing calculator - piece of cake. At least for a decimal approximation.  Here it is
on Desmos, along with Khayyam's geometric solution.

The last thing I want to do is to verify that his solution works. The question of how did he derive this is important, but I'm not going to get to it here.

So the equation of the circle is \( (x-b/2a)^2+y^2=(b/a)^2 \) and the parabola \( y=x^2/ \sqrt(a) \). So... hmm. Substituting the parabola equation into the circle gets us a quartic!

So why a circle? It gets us a right triangle, which gives us proportions.
\[ \frac{x}{(x^2/ \sqrt{a})} = \frac{x^2/ \sqrt {a}}{b/a - x} \\
\frac{\sqrt {a}}{x} = \frac{x^2/ \sqrt {a}}{b/a - x} \\
a(b/a-x) = x^3 \\
b - ax = x^3 \\
b = x^3 + ax \]


(The mathjax is displaying odd for me, \( \sqrt a \) is square root.)

I did think it was interesting that none of the students had any idea how Khayyam was drawing parabolas before graphing. Maybe we'll have to do some directrix learning.

p.s. Deborah Kent and Milan Sherman (Drake University) wrote a great extended piece on this.

Monday, May 23, 2016

More Tessellations

In class we talk about Islamic tessellation art, going from this Google doc with images and resources. (The student's Google doc is editable, and they add their work to it. The full class one is 30 pages now of previous classes work!) Then the students get free time with pattern blocks, grid paper and some online applets.

Some of my students' work. With a few of my attempts at think further questions...

 Erin blogged about this, too.

 Anthony made this on triangular grid paper. I think I wished he differentiated the cubes a bit - but it's his vision!

 Very interesting tessellation from Becca. She made this cool piece and then saw what she could make from it. In our Facebook group I asked "So the fundamental domain is the smallest piece that repeats to fill the page. What might that be here?"

Marty made these two. The hexagonal symmetry is a powerful pull with pattern blocks.

 Is Heather's hexagonal? On FB I asked: "Hmmm... so what's the ratio of red to blue if you go on to fill the whole plane?"

I thought there were interesting connections between Brianna's pattern blocks and Nick's triangular grid design.

On FB I asked about Brianna's: "Another good one for the fundamental domain question: what's the smallest set of blocks that you could repeat to make the whole pattern? I think this one would include parts of blocks!"

One more from in class: Andrew made overlapping triangles and was seeing what would happen.

Some work shared on the tessellation page:

Some Math Toybox creations from Hannah.


Jordan's triangular grid creation and Kourtney's square grid.

A few more rolling in:
Tabatha's online tessellation - nice detail work around the red vertices.

Nick's (the other one) square grid. He notes "I did try to get alot going in this. all of the negative space (not colored in) creates triangles. each center piece, negative space square behind center pieces, the Octagonal shapes around the negative squares, all rotate from left to right, clockwise. all of the corner pieces reflect from column to column horizontally and vertically. Embarrassing enough it took me a long to make this work. The initial "pattern" i thought of like a stamping, which would be the top and bottom row. then I worked inwards."

And we'll finish with Andrew's finished work. I like the Archimedean polygonal spiral... might have to make something like that.


Sunday, May 22, 2016


Daniel Mentrard is one of my GeoGebra heroes. The guy can seemingly do anything, and he is generous with help for his lessors.

Recently he posted this:

Don't stare too long - this is hypnotoad territory.

I misunderstood it. I thought it was a single triangular tile, sometimes flipped and sometimes rotated. So I got playing around with it, trying to generalize to other cases than the hexagonal. I made a simple triangle, with two triangles in it. The red has vertices on the border, and the blue has vertices in the region. (The commands are Point[<object>] for the first, and PointIn[<region>] for the second.) The other command is Sequence. The basic structure is usually something like Sequence[Rotate[thing, i*360o/n, centerpt],i,1,n].

GeoGebraGeek paragraph: The fine point in this one is I wanted it do do something differently for 2, 4, 6 than for 8, 10, 12. So I defined a Boolean variable, a = n >7. Booleans read as both true/false and 1/0. The numeric is handy for dynamic coloring (eg. turn something green if it's where you want it) or for goofy stuff like this; Sequence[Rotate[thing, i*(1+a)*360o/n, centerpt],i,1,n/(1+a)]. That makes it skip every other side for the values where they'd overlap.

Then some motions and - voilà - you get a kind of tiling.
I like the hexagonal and the square symmetries the best. Though that dodecagonal could be neat with a complementary triangle tile for the gaps.

With World Tessellation Day coming up, I've been thinking about tessellations a lot. I've mostly concentrated on Escher style, but I think I want to look more into decorating plainer tiles and then seeing the results of the symmetries, too.

The sketch is on for play, of course. (Some of my other tessellations are in this geogebrabook.)

Now I'm thinking one of the decorations should have been a quadrilateral. Next time!

A gif of using the sketch; A decagonal symmetry.

Square symmetry and dodecagonal for the same decorations.

Saturday, May 21, 2016


Just a cartoon.

With love for all the great WODB out there.

And a HT for @mathtans.

Friday, May 20, 2016

Learning Slow

My spouse and I have been learning Tai Chi with a seniors' class, and it is awesome.

We've been going for maybe 15 months, and we haven't finished the first form yet. (Yang 24)

Marna is the teacher, and she is great. She knows her students, what they need, and how to help them reach it. She does lots of formative assessment, watching us, trying just hands and just feet, sometimes having us watch her. If a student forgets, it's okay. If a student is new, "just fudge it until you understand." She lets us know the benefits of what we are doing. She shares with us stories from when she learned from Paul Lam. She knows that doing what we're doing is the point. Even if some students really want to know and keep the form.

We had a short break while she had two knees replaced, and then she was back teaching exercise classes over two counties.

I'm learning patience as a learner. I hope I'm learning patience as a teacher.

As well as to focus on what doing math is doing for my students.

Let's breathe, shall we?

p.s. I know the title sounds like bad grammar, but it's what I mean.