Problems: special right triangles

I've added another problem to the Problems page.

Using your knowledge of 30-60-90 right triangles, prove that the area of an equilateral triangle with side of length $latex s$ is given by $latex A=\frac{\sqrt{3}}{4}s^2$. Use this formula to find the area of a regular hexagon with sides of length $latex s$.

Odds and Ends: April 19, 2013

Two items that might interest you:

First, I mentioned online LaTeX compilers in this post. Those compilers are great for math teachers (or anyone) who uses multiple computers, especially computers where they lack administrator rights. A year later and I've come across a new online LaTeX website  which looks better than anything else. It's called writeLaTeX and there are many reasons you really should take a look:

  1. It's free! Just go there and start typing.
  2. It's simple to use. The diagram says it all.
  3. There are templates provided for creating a paper, presentation, and more!
  4. As you type, you're given a "rapid" preview of the changing document.
  5. If you sign up for writeLaTeX then you can collaborate with others on LaTeX documents.

When you go to the website you'll see 2 buttons: one to create a new paper and the other to create a new presentation. But if you look a little further down there are links for a "WriteLaTeX Tutorial", "Templates and Demos", and "Sign up for a free account". Clicking on "Templates and Demos" will take you here, where you'll find you actually have access to 3 pages of templates. Even better, writeLaTeX is available for your I-pad, tablet, and mobile device.

I spent some time experimenting with it and found that the "rapid preview" was not so rapid on my machine. Gummi, the IDE I use in creating many of my posts, updates much more rapidly than writeLaTeX. This, however, I think is a minor issue. writeLaTeX looks to be a useful tool every math teacher should be aware of. I've added a link to writeLaTeX on the sidebar.

Second, there is a good article on "Academias Indentured Servants". Though it isn't directly related to high school math, it does relate to an earlier post where I mentioned the pool of qualified people who could teach at the high school level if it weren't for the certification requirements. Of course, not every adjunct professor would be suitable to teach at the high school level (eg failure to pass a background check) nor would some want to teach at the high school level. But, having seen this situation up close, I also know there would be a decent percent that would abandon the poverty wages of academia for a high school teaching job--if only those onerous certification requirements were removed.

Altermundus' packages

EuclideLinesThe English version of the Altermundus site showcases the beautiful work of the author, Alain Matthes, using packages that he created. They're built on PGF and Tikz and, like the sagetex package, are some of the most important packages a math teacher should learn---especially the tkz-euclide package for geometrical drawings. Since the packages are built on Tikz it's possible to use Tikz code, too. The problem (at least for me) is that the documentation is in French which makes going through it a bit slower than I'd like.

I've set up a new page to help me learn the packages and I'm starting with tkz-euclide. It's organized differently than the actual documentation and has simple code for you to download and experiment. There's a link to Altermundus' packages on the LaTeX page, the sidebar, or you can click here.

Sage Interact: trigonometric transformations

SinCosTanThe screenshot above illustrates a new manipulative I've put together, running on the Sage Sandbox page, that let's user explore transformations of the trig functions sine, cosine, and tangent. More specifically, the user can experiment with how the values of A, B, C, and D affect the graph of:

  1. y=Asin(Bx+C)+D
  2. y=Acos(Bx+C)+D
  3. y=Atan(Bx+C)+D

Two issues came up along the way:

  1. How do you control the range of y-values in the output?
  2. How do you make tangent plot as a disconnected graph?

I've added information to address these topics to the Sage Interact Essentials page. The code for the manipulative can be found on the Python/Sage page.

"Proof": the real numbers are uncountable

In an earlier post, I gave my opinion that the most important lesson of high school math should be that the rational numbers have measure 0. As part of that mission, the rational numbers were ordered in this post. Today I've added the "proof" that the real numbers aren't countable to the Handouts page. No doubt some of you have seen it before but it's rot really a proof unless you've established when two different decimals represent the same real number. Clarifying that issue, unfortunately, makes a real proof too difficult so you're best off asserting that mathematicians have proven this to be true. I've given a sketch as to why that's so as well in case you get asked. By focusing on the "proof" (rather than proof) that the real numbers are uncountable it's within the grasp of an accelerated math class.