Sage Interact: Area of a Circle

Before the formal concept of a limit in Calculus, students get a chance to see informal limit arguments in earlier classes. For geometry, I'm supposed to explain how students can deduce the area of a circle by breaking it up into many sectors. It's a very well known argument so I went looking around the internet for resources. I found what I wanted here; it's just one of many useful Geogebra demonstrations that a math teacher might need. If you teach math then the site deserves a closer examination.

But for me, having a Sage manipulative is better because I can modify the code to make it look the way you want. Notice that there are always an even number of sectors so the shape is more like a rectangle/parallelogram rather than a trapezoid. I decided I wanted to use 2 different colors which I would alternate when pulling the circle apart. This will make it clearer that the base of the resulting object ("close" to a rectangle or parallelogram) has length (circumference)/2. The code is available on the Python/Sage page and here's a picture of it running in the Sage Cell Server.


I also put a brief explanation into the code for the benefit of my class. The graphics (no text included) can be downloaded  by clicking on the "AreaOfCircle.pdf" in the lower lefthand corner. I've posted 7 such examples (using a different number of sectors) on the Graphics page.

Here's a picture of it with 40 sectors in the circle:


As the number of sectors increases the figure on the right becomes more and more like a rectangle with base $latex \frac{C}{2}=\frac{2\pi r}{2}=\pi r$ and height $latex r$. This area could be approximated with the area of a rectangle: $latex A=\pi rr=\pi r^2$. The area of the circle and "rectangle" will have the same area, so the area of a circle is $latex A=\pi r^2$.

Chess Basics: The Pin

In chess, a piece is pinned if moving it exposes a more valuable piece to attack. If the piece is pinned to the king then the laws of chess make it illegal to move the pinned piece; the king can never be exposed to attack. That type of pin is called an absolute pin. If the laws of chess allow the pinned piece to move then it's a relative pin. You'll need to be on guard and consider the consequences of what happens if the pinned piece moves. I've posted a piece about the basics of pins on the Chess page.

Sage Interact: H-Trees

I've been really busy lately as winter break approaches, so I haven't had much free time. This past weekend, though, things finally finally slowed down so I was finally able to check in at my favorite sites. is one of those sites. There's been a lot of interesting additions but the one that intrigued me most was on H-Trees (and b-trees). $latex \LaTeX$ is amazing but when it comes to real math I have to us Sage; that's especially true when it comes to fractals because the power of a CAS is almost certain to produce more iterations of the fractal. More iterations is better!

So I created a Sage Interact manipulative to create H-Trees. The code is posted on the Python/Sage page and the graphical output of iterations 1 through 6 are posted on the Graphics page. The code is set for 4 iterations because I wasn't sure how much trouble your computer would have producing more iterations. In order to get more iterations I had to define multiple Graphics() objects but even then I wasn't able to produce many iterations on Sage 4.8. I had better luck with running it on the Sage Cell Server. Here it's running 6 iterations; this screenshot is posted on the Sage Interact: Fractals page, too.


Notice there's a link at the bottom which says "H-Tree.pdf"; you can download the PDF output by right clicking on the link and saving it to your computer. Note: It looks better than the screenshot indicates. I needed a small figure size to get the code into the picture.

After some experimenting I decided to make the later iterations have thinner lines, as was done on That code is also posted on the Python/Sage page. A small number of iterations didn't look very good because the varying widths results in the some of the lines visibly "overshooting" the connection. With 6 iterations though, that drawback isn't as blatant. Here's the result, the PDF is posted on the Graphics page.


$latex \LaTeX$ is great but for some tasks but Sage is better! Yet another fractal you can use in the classroom.

More Links: 12/15/12

I've added 2 more links to the sidebar:

Salvus is the blog for William Stein and, according to his blog, "...Salvus project as a successor to the free public Sage Notebook server"..." and " The primary goal is to make Sage and other sophisticated free open source mathematical software available to a large number of simultaneous users.".

Interact Sage is, from the FAQ section: "...a public space for Sage users to interact; it makes use of the Sage cells to allow Sage users to easily share snippets of code with each other. Users can browse others' code snippets, add their own, bookmark their favorites, and more."

Make sure you check out this example from the Interact Sage site; very cool!

London Chess Classic 2012

The London Chess Classic 2012 has just ended and the result is no surprise: Carlsen wins again. He had, after all, already won the tournament before the final round began on his way to a 2994 performance rating. Along the way Magnus got some attention for having the highest FIDE live rating ever, eclipsing the legendary Gary Kasparov. It also puts him more than 50 points over the world number 2. If it weren't for Magnus' incredible result, the star of  the show would have been Vladimir Kramnik; he's vaulted into the number 2 spot in the world by impressive play against this very strong field.

But for me, the interesting side story is the continued lackluster performance of Vishy Anand. One of the greatest players to ever play the game, Vishy seems to be unable to beat the best in the world anymore; that's not what you expect from a world champion. As is mentioned in this post, a lot of top players have pointed out Anand's game is in decline. This tournament is more evidence that Anand isn't going to get back to the elite 2800 club any time soon.

As a result, Anand's rating and ranking continue to drop. In November of 2010 he was ranked number 1 in the world, with a rating of 2804 but there has been only one direction over the last 2 years: down. At the end of 2011 he was at number 4 in the world, after the most recent world championship win he was at number 6 in the world at 2780. Now, near the end of 2012 the live ratings have him ranked 7 at 2772.1, just barely ahead of Topolov. Even more startling, he's about 90 points below Carlsen, a huge amount at this level. This underscores that Anand is world champion in name only. But time is not on his side: you don't hit your peak rating at 42 years of age. Chess is a young man's game. Notice that Anand had "just" a 2749 performance rating and his only win of the tournament was versus Gawain Jones, the cellar dweller of the tournament with a rating closer to 2600 than 2700.

But don't forget about Anand's foe for the recent world championship, Boris Gelfand. He  recently finished his own tournament: Tashkent 2012 and finished 10th (out of 12) with no victories and 2 defeats, making it clear (I hope) that Gelfand doesn't belong in the top 10, even though he did just compete in a world championship match. The latest tournaments are more evidence that Kasparov was right.

1 divided by 0 is undefined

From my experience, a lot of high school students are under the impression that $latex \frac{1}{0}=0$. Here's a brief argument to show them why that's wrong; it's a good example to use if you teach proofs, too.

$latex \frac{6}{3}=2$ because $latex (3)(2)=6$

$latex \frac{15}{3}=5$ because $latex (3)(5)=15$

$latex \frac{200}{25}=8$ because $latex (25)(8)=200$

$latex \frac{7}{8}=0.875$ because $latex (8)(0.875)=7$

This relationship will hold between any any three numbers involved in division. Therefore,
if $latex \frac{1}{0}$ was defined, say $latex \frac{1}{0} = k$, then $latex 0(k)=1$. Since $latex 0(k)$ must equal $latex 0$, for any number $latex k$, the assumption that $latex \frac{1}{0}$ was defined is false.

You can find the tex file and PDF output on the Handouts page.


A Glimpse of Calculus

Calculus has a lot of information to cover; too much for the typical high school student. I combined introducing a graphing calculator, the concept of a good graph, and even a glimpse of calculus in my precalculus course. That helps get you through some calculus concepts a little bit quicker.

Here's what you do:

  1. take optimization problems from a calculus book and spend some time having students formulate the function to be graphed (weak classes can be given the formulation but that's one more thing to teach in calculus)
  2. graph the function using with a graphing calculator
  3. estimate the relevant minimum or maximum using the graphing calculator.

That's how the graphics from the previous post came about: using formulas for the surface area and volume of a right circular cylinder you can derive the function describing the surface area of the paint can as a function of the radius (in inches). And now you've got the chance to promote the power of calculus: technology can help us estimate the minimum/maximum but calculus will let us determine the precise value of the minimum/maximum. When you teach calculus you can revisit 1 or 2 of the problems to determine what the exact answer is.

I've typed up the example that creates the graphics below. The PDF is on the Handouts page and if you want to compile the tex source file then you'll  need the PDF of the Paint Can Curve on the Graphics page.

Odds and Ends (12/2/12)

First, the Graphics page has a new addition: a tex file and PDF file to create the following graph.

The graph shows the surface area of a paint can as a function of the radius. I'll use it in an upcoming post.

Second, there is an interesting article on best educational systems in the world. A new report by Pearson called The Learning Curve confirms what other studies have repeatedly found: the US is about average for high school math. The top 5 were:

  1. Finland
  2. South Korea
  3. Hong Kong
  4. Japan
  5. Singapore

while the US came in at number 17. The report notes that Finland and South Korea were 2 very successful systems that are dramatically different and they conclude:  "while funding is an important factor in strong education systems, cultures supportive of learning is even more critical -- as evidenced by the highly ranked Asian countries, where education is highly valued and parents have grand expectation. While Finland and South Korea differ greatly in methods of teaching and learning, they hold the top spots because of a shared social belief in the importance of education and its "underlying moral purpose."".

The article indicates mentions an alarming statistic: "Just 6 percent of U.S. students performed at the advanced level on an international exam administered in 56 countries in 2006. That proportion is lower than those achieved by students in 30 other countries.".