LaTeX: (Big) Scalable Fonts

Although $latex \LaTeX$ gives you some choices on the changing the default font size, there will be times when you want something bigger. And whenever you find yourself thinking, "It would be really great if...", chances are someone has designed a package that will allow you to do what you want. There are several packages for scaling the fontsize well beyond anything available in $latex \LaTeX$. I've put together a template using the fix-cm package and added it, along with some basic instructions, on the ...change the default fontsize? page. Here's a screenshot:

BIGfonts

There are 2 issues to be aware of. The first is when you use a big font combined with small text (such as through \tiny ); the fonts were designed to be typeset as \tiny but now \tiny is still big. As a result, the text might not look as well with the fix-cm package. The second issue is the spacing between the lines. There's a rough guideline that your baseline distance should be 1.2 times the font size for things to look pleasing to the eye. So if your font size is 30 pt then the distance between the baselines is about 36 pt. These guidelines aren't rules. Ultimately you'll have to decide what works with best.

Still interested? Then click here.

Tata Steel Chess Championship 2013

The Tata Steel Chess Tournament 2013 has ended and it's deja vu all over again: Carlsen wins (like the London Chess Classic 2012) before the final round was played. This pushes his live chess rating up more than 11 points to 2872.2, even further into realm of chess god and making the thought of the first 2900 chess player a distinct possibility. His crushing win over Nakamura was my personal favorite.

Second place belonged to the second seed of the tournament, Lev Aronian, losing just one game (to Anand). This was a solid result that could have won the tournament but unless Carlsen has a poor tournament it's not good enough: Carlsen was comfortably ahead by 1.5 points. Vishy Anand took 3rd place on tiebreaks over Karjakin. It was a good result for him, too, as he halted his recent slide picking up 7 points for the tournament and increasing his ranking 1 spot. His demolition of Aronian was considered by Anand to be one of his best games: with a phenomenal career such as Anand, that's saying a lot. Anand's performance looks a little bit shakier when you consider that the win was the result of home preparation for the Gelfand match. Add in the observation that no one in the  top 6 except Anand lost to a player who ended up 4th or lower. Next notice that Hou Yifan was able to get a draw against him (but none of the others in the top 4). Finally, the loss in the final game to Wang Hao kept him from 2nd place. But all in all, a solid tournament for Anand.

The surprise, for me at least, was Hou Yifan who fought so well. Despite being the lowest rated of the tournament she managed 3 wins and was able to nick Anand for a draw. She ended up gaining more points from the tournament than any other player. For a player with such a long career ahead of her she is poised to be the greatest woman to ever play chess. Only time will tell.

How do I...add graphics and text to a picture?

Tex Stack exchange, located on the sidebar, is the place to go if you want to learn about LaTeX or solve some problem in LaTeX. One of the the most useful and popular posts was this one, about writing on a picture using the tikz package. I've taken the basic information there and (hopefully) simplified the process in the latest How Do I...? topic from the LaTeX page.

The new page has a tex template for you to download; just follow the directions using your picture and you should be able to get graphics and text added to your picture in no time.

WriteOnPic4

Sage Interact: Dragon Curve

Yet another fractal for your class; this one is known as the Dragon Curve; more specifically the Heighway Dragon. It begins with a single line segment. Call one endpoint START and the other END. Rotate the curve 90 degrees about the END and then superimpose it with the original path to create a curve with 2 line segments. The START hasn't changed but the END has. Rotate the new curve about the new END and superimpose it on the curve of 2 segments to create a curve with 4 segments. Continue the same process. Here's what it looks like after 12 iterations, running in the Sage Cell Server:

DragonCurve12After 18 iterations

DragonCurve18iter

The code for the Dragon Curve is on the Python/Sage page. Remember, all you have to do is copy the code and paste it into the Sage Cell Server to have your own manipulative to experiment with. I've put the PDF of the output for 12 and 18 iterations on the Graphics page. The picture of 18 iterations is posted on the Sage Interact: Fractals page.

Proof: the Zero Property

I've added a proof of the Zero Property to the Handouts page as a PDF and as a tex file. The Zero Property says if $latex ab=0$ then either a or b must be equal to 0. Students use this when they solve for roots of polynomials: If $latex x^2-1=0$ then $latex (x-1)(x+1)=0$ and the Zero Property tells us either $latex x-1=0$ or $latex x+1=0$ from which we conclude $latex x=-1$ or $latex x=1$. Besides being relevant to high school math, the proof is so simple it should be a standard example in the high school math curriculum.

Problem: Combinatorics and Inclusion/Exclusion

Here's a problem that deserves a spot on any exam covering the basics of combinatorics. Start with 2 prime numbers, for example, 3 and 11 and one "large" number, for example 15000.

How many numbers between 1 and 15000, inclusive, are divisible by 3 or 11?

By adjusting how large the number is you can prevent students from trying to count their way to an answer. Moreover, changing the 3 numbers will allow you to create many similar problems.  The problem forces the student to count the numbers divisible by 3 along with those divisible by 11 and then discard the numbers divisible by both (ie divisible by 33) because they've been counted twice. That's the easy case of a more general result known as the Principle of Inclusion/Exclusion; if your class had to learn it then the problem can be extended to count how many numbers are divisible by at least one of 3 different primes.

The fact that this problem can be modified so easily makes it one of my favorites. On a personal level I like to start out the "discussion" with a difficult version such as "How many numbers between 1 and 1,000,000 (inclusive) are divisible by either 2, 3 or 7?". Nobody ever raises their hand which allows me to drop back and say, "By the end of this year you'll be able to do problems even harder than this.". If you start with smaller numbers and a simpler problem such as: "How many numbers between 1 and 10 are divisible by 3?". "How many numbers between 1 and 10 are divisible by 2?". "How many numbers between 1 and 10 are divisible by 2 or 3?" then with time and practice any student who is proficient in division can master it. Students can get a sense of accomplishment that a problem which seemed so impossible now is kind of easy.

I've posted the problem on the Problems page.

Sage Interact: Newton's Method Fractal

During the all-too-short Christmas break I found some time to play around with Sage. I don't know that I could use this example in the classroom, but maybe you've got some ideas. After all, if you teach Calculus you might have the opportunity to teach Newton's Method. Newton's Method can be extended to the complex plane, and the various roots act as points of attraction as the iterations increase. The link above shows the basins of attraction for $latex z^5-1=0$ but I went for $latex z^3-1=0$ to get this well known fractal.

NewtonsMethod

The image above is created using 250 by 250 points (62,500 points) in the complex plane. Each point is iterated through Newton's Method 5 times; it took more than 5 minutes of computer time.

Decreasing the figure size will get the "pixels" closer together (which makes the image darker). I've posted the PDF output for figure size 7 and 8 on the Graphics page. Each image is about 10 megabytes so, depending on your download speed, that might take awhile.

I've also posted the code on the Python/Sage page. I dropped the max iterations down to 3 so you'll need to adjust that higher to get the picture I created. Starting out with a smaller size problem will let you see how long it takes on your computer.

How Do I....change the page style (number/header/footer)?

I've added another topic to the How Do I.... section of the LaTeX page. The page addresses how to control the page numbering as well as the header and footer of your $latex \LaTeX$ document. By default, you get page numbers in the center of the footer of each page. This new page will explain the basics of turning the page number off, putting the page number in a different place, adding a decorative line for the header or footer, adjusting the thickness of the line, putting a picture into the header/footer and more. Here's a screenshot of the tex code running in Gummi: note the page number in the upper right hand corner and the picture in the lower left hand corner.

PageStyle

You can go to the Latex page to access the new page but it's also on the sidebar. You can  click here as well.