Glen Whitney @ RM!

Posted on October 14, 2010 by Mr. Chase

As I advertised over a month ago, Glen Whitney will be joining us on Friday, October 22 for a walking math tour of Rockville. I’m excited to hear what he has to say (and to steal all his good ideas!). If you’re a Calculus student, or if you’ve taken it, and you’re not yet signed up for the field trip, stop by my room and we’ll talk. There are currently four slots left.

Reposting some of the info:

Richard Montgomery High School will be hosting mathematician Glen Whitney on Friday, October 22. He will be doing a walking ‘math tour’ of downtown Rockville.

Glen is the executive director of the Mathematics Museum on Long Island NY. For more information about Glen, or about the math museum, here’s an article about him. Or, visit the math museum’s website.

And here’s the article that’s on the RM website.

Home, Home on the Range

Posted on October 13, 2010 by Mr. Chase

The first week of school I gave this problem and never came back to it:

Find the range of $f(x)=2^{x^2-4x+1}$ .

The answer is $\left[\frac{1}{8},\infty \right)$ . Here’s why:

First consider the range of the quadratic in the exponent, $h(x)=x^2-4x+1$ . It’s a parabola that opens up with its vertex at $(2,-3)$ . So the range of $h(x)$ is $[-3,\infty)$ .

Now, consider the function $g(x)=2^x$ . If we let the domain of $g$ be the values coming from the range of $h$ , we have the mapping $\left[-3,\infty\right)\to\mathbb{R}$ . That is, we’re considering the composition $f(x)=g(h(x))$ . Since $g$ is monotonically increasing, for any $x_1<x_2$ , we know $g(x_1)<g(x_2)$ . So the range of $g$ in $\mathbb{R}$ is $\left[g(-3),\infty\right)$ . So the range of $f$ is $\left[\frac{1}{8},\infty \right)$ .

Do you feel “at home on the range”?

Here are a few more for you to try. In each case, find the range of the function. These aren’t meant to be any harder than the original problem, just different. Though watch out for the third one :-).

$y=3^{-x^2+4}$

$y=4^{\sqrt{x}}$

$y=\ln{(x^2-x+1)}$

$y=\sin{(x^2+2x+5)}$

Powerful Problem (hint)

Posted on October 12, 2010 by Mr. Chase

A few weeks ago I posted this “powerful” problem:

Solve $\left(x^2-5x+5\right)^{\left(x^2-9x+20\right)}=1$

Now, allow me to give you a major hint. Consider the simpler equation

$a^b=1$

What are the possible values of $a$ and $b$ ? Here are the possible combinations:

$a=1$ and $b$ is anything
$b=0$ and $a$ is any nonzero number

And here’s the tricky one that most people forget:

$a=-1$ and $b$ is even

You now have enough information to solve the original equation. I think you’ll be delighted with the solution!

10/10/10 @ 10:10

Posted on October 10, 2010 by Mr. Chase

Since this is a math blog, I felt obliged to capitalize on today’s date. I’m sure everyone has noticed, like they do every year when 9/9/9 comes up or whatever else. But today is a bit more special, since 10 is our favorite base to work in. To celebrate powers of 10, head over to the Let’s Play Math blog to see lots of links and a video that will put you in the mood :-). Happy day!

Equations from photographs with new Casio calculator

Posted on October 9, 2010 by Mr. Chase

Excerpt from wired.com’s gadget lab:

If you’ve ever looked at the curve of a hill, the cables of a suspension bridge or the arc of a coastline and wondered, “I wonder what function would fit that line?” — congratulations, you’re a nerd. And Casio has a surprising new calculator that will answer your question.

Casio’s new Prizm calculator is to the graphing calculators of my school-days as the iPad is to the slates we scratched on with sticks of chalk. It has a color, 216×384 pixel display, 16MB of memory, a USB-port, and will do all of your math homework for you….

Any graphing calculator will let you input an equation and show you the result. Casio’s Prizm does this in reverse. The color screen will display a picture, and will draw a line over the top of any shape you like. It will then give you an equation for this line.

If that wasn’t amazing enough, that USB port lets you hook the calc up to a compatible Casio projector to show off the results on the big screen….

$130, available now.

That’s pretty amazing. I’m not sure it’s the most important feature of this new calculator, but it does highlight the fact that our TI-83’s haven’t changed significantly in 14 years. It’s surprising, given the advances in every other area of technology. Why isn’t your TI-83 calculator three or four times smaller and 20 times more powerful than the TI-83 of 14 years ago?

Let’s boycott Pizza Hut!

Posted on October 9, 2010 by Mr. Chase

I’m kind of kidding (but only kind of!). Can you believe they made this commercial? Unbelievable.

Apparently it’s been pulled off the air, so I think Pizza Hut is somewhat penitent for what they did. Good thing.

Irrationals of the form a+b√c

Posted on October 8, 2010 by Mr. Chase

I made the claim in a post last week that the set of irrationals of the form $a+b\sqrt{c}$ is countable. I said that pretty quickly, without justification. I’ve never proved statements like this before, but here I’m going to try.

Theorem. The set of irrationals of the form $a+b\sqrt{c}$ , with $a,b \neq 0,c>0\in\mathbb{Q}$ , is countable.

Proof. Consider the set of irrationals of the form $a+b\sqrt{c}$ , with $a,b \neq 0,c >0\in\mathbb{Q}$ . More formally, define

$\mathbb{I}=\left\{a+b\sqrt{c}\in\mathbb{R}: a,b \neq 0,c>0\in\mathbb{Q}\right\}$

And also require that $c$ is ‘square free’–that is, we require that neither the numerator or denominator of $c$ contain factors that are perfect squares. So $a+b\sqrt{c}$ is in ‘simplest’ form. We aim to show that $\mathbb{I}$ is countable.

Now, consider the function

$\mathbb{I} \overset{f}{\rightarrow} \mathbb{Q}^3$

defined

$f(a+b\sqrt{c})=(a,b,c)$

This function is one-to-one since we require $c$ to be in simplest form–that is, the image of any number $a+b \sqrt{c}$ under $f$ is unique. So $f$ is an injection from $\mathbb{I}$ into $\mathbb{Q}^3$ .

We know that $\mathbb{Q}$ is countable. Since a finite Cartesian product of countable sets is countable, $\mathbb{Q}^3$ must also be countable. And we have constructed a function $f$ which is an injection from $\mathbb{I}$ into $\mathbb{Q}^3$ . So the cardinality of $\mathbb{I}$ must be no greater than the cardinality of $\mathbb{Q}^3$ . Thus $\mathbb{I}$ must also be countable, as desired.

I think I did that right. Any suggestions, math readers?

Self-organizing Classrooms

Posted on October 8, 2010 by Mr. Chase

I asked recently if your (math) teacher could be replaced by video. I was kind of serious. Now, I ask, could your teacher be replaced by the internet?

This TED talk is fascinating. For my students reading this, would you like to learn this way? In some ways, you probably already do. What would you think if I just let you loose and said, “Here are the objectives of the course. Go learn them.”? In what ways would that be better or worse than what I’m doing now?

The Extended Bell Curve

Posted on October 7, 2010 by Mr. Chase

Today’s GraphJam. This goes out to my Precalculus class from last year–especially one of you. You know who you are!

One might say that this dinosaur is a ‘head above the rest’… (groan :-)).

Google awards $2 million to AIMS

Posted on October 6, 2010 by Mr. Chase

News from the TED Prize blog:

Congratulations to the African Institute for Mathematical Sciences (AIMS) and 2008 TED Prize winner Neil Turok on winning $2 million in funding from Google’s Project 10^100. Project 10^100 (10 to the 100th power) was a call for ideas to divide a $10 million fund into five pieces that would help as many people as possible around the world. AIMS’ piece of the prize will be used to expand its network of science and math academies that promote graduate-level study in Africa.

Interestingly, Khan Academy, which I raved about in this post, has also been awarded a TED Prize. Click here for more info, and a nice video about it (sorry the news is a bit old!).

Random Walks

Mr. Chase blogs about math