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On Christmas eve my family and I went around looking at Christmas lights. This is the Balian House in Altadena. We used to go and look at the Christmas lights here when I was much younger and it’s actually exactly as I remember. Apparently the Balian family used to (or still does) own an ice cream company and this was the house they lived in. I’m not sure if anyone lives here now. I think the family just owns it and they still decorate it for Christmas.

The Balian House

The day after Christmas my family went shopping, I decided to go to the Huntington Library in San Marino. It was very crowded. I especially like the Japanese garden they have.

Japanese Garden

Zen Garden

So this is Christmas
And what have you done
Another year over
And a new one just begun
And so this is Christmas
I hope you have fun
The near and the dear one
The old and the young

A very merry Christmas
And a happy New Year
Let’s hope it’s a good one
Without any fear

And so this is Christmas
For weak and for strong
For rich and the poor ones
The world is so wrong
And so happy Christmas
For black and for white
For yellow and red ones
Let’s stop all the fight

A very merry Christmas
And a happy New Year
Let’s hope it’s a good one
Without any fear

And so this is Christmas
And what have we done
Another year over
A new one just begun
And so happy Christmas
We hope you have fun
The near and the dear one
The old and the young

A very merry Christmas
And a happy New Year
Let’s hope it’s a good one
Without any fear

War is over, if you want it
War is over now…

Happy Christmas

Now that I have a little bit of spare time, I’ve been able to catch up on some reading. I’ve been meaning to read the books listed bellow for a long time. I’m glad I finally got around to it.

The Last Lecture by Randy Pausch

Randy Pausch’s story is truly inspiring. I’m sure many are already familiar with it. Randy was a computer science professor at Carnegie Mellon University. In 2006 he was diagnosed with pancreatic cancer and in August 2007 he was given a terminal diagnosis: “three to six months months of good health left.” He gave his “last lecture”, titled Really Achieving Your Childhood Dreams, in September 2007 to an audience of about 400 people at Carnegie Mellon.

This book is full of stories from Randy’s life. I particularly like this one. Randy had just created a new course called “Building Virtual Worlds”. He gave the class its first two-week assignment and was blown away with the results. He called his mentor Andy van Dam. “Andy, I just gave my students a two-week assignment and they came back and did stuff that, had I given them an entire semester to complete it, I would have given them all A’s. What do I do?” Andy’s reply, “OK. Here’s what you do. Go back into that class tomorrow, look them in the eyes and say ‘Guys, that was pretty good, but I know you can do better.’ ” Randy followed his advice and the students continued to impress. Enabling the dreams of others was a main theme in Randy’s life and this story demonstrates one way he was able to do that. Never underestimate people, let them fulfill their potential.

Three Cups of Tea by Greg Mortenson

Another really inspiring story which I’m sure many are already familiar with. While attempting to climb K2, Greg and and three other climbers had their ascent interrupted by the need to rescue of a fifth climber. After getting lost during his descent, Greg takes refuge in the small village of Korphe. Inspired by the their hospitality and shocked by their lack of education, Greg promises to repay the village by building a school.

The book chronicles Greg’s struggles as he attempts to build a school for Korphe. Starting from literally nothing, Greg is able to raise money and purchase supplies for the school. Greg’s work doesn’t go unnoticed, after many hardships people in the US and Pakistan start to see the good that he is doing. With the help of a wealthy philanthropist, Greg becomes a founding board member of the Central Asia Institute (CAI). The CAI’s mission is to build schools and promote education throughout Pakistan. Greg’s story is proof that one man can really make a difference in the world. Although initially met with some resistance, it is amazing how much support he has gotten.

I still have a couple of weeks left on my break. I’ve heard Water for Elephants is good, perhaps I’ll read that. Any other suggestions? NPR has a selection of best books of 2010 here.

The word (almost) every math student hates, “proof”. When teaching a precalculus class I do not introduce the notion of a rigorous mathematical proof, however, I do try to get the students to understand why a particular result is true. I use the phrase “why might this be true” a lot. My goal is to get the class thinking about how we arrive at such a conclusion instead of merely accepting it. This doesn’t always work, in fact it rarely does. Students are too content with just memorizing a formula and knowing how to use it. If it work, it works, who cares why it works. To a certain extent that’s fine, I mean, it is important to know how to apply a result. But my point is this, it’s just as important to ask why something is true and at least try to understand why it is true. Even if you don’t get it completely.

To emphasize my point, I’m trying something new for my class next quarter. In the past when I’ve taught the properties of logarithms, I would simply write a property on the board followed by a brief justification. I had fallen into the mindset of my students. I didn’t care if they knew how to derive the properties of logarithms, I just cared that they could memorize them and know how to use them when solving equations. I know, I’m a hypocrite. For next quarter though, I’ve come up with a worksheet that will combat this.

Section 5.4 Properties of Logarithms

In this worksheet we will derive some important properties of logarithms. These properties will be essential for us when we solve logarithmic and exponential equaitons in section 5.5.

Before we can talk about the properties of logarithms we need to recall a few important facts.

• The definition of a logarithm: $\log_b(x)=y$ is equivalent to $b^y=x$.
• One to one property of logarithms: If $\log_b(x)=\log_b(y)$ then $x=y$.
• One to one property of exponents: If $b^x=b^y$ then $x=y$.

The Inverse Rules

$\log_b(b^x)=\bigstar$

What might $\bigstar$ be? First use the definition of logs to rewrite as an equivalent exponential equation. Then use the one to one property of exponents to find $\bigstar$.

$b^{\log_b(x)}=\bigstar$

What might $\bigstar$ be? First use the definition of logs to rewrite as an equivalent logarithmic equation. Then use the one to one property of logs to find $\bigstar$.

The Logarithm of a Product

$\log_b(MN)=$ ?

To derive a formula we make a substitution. Let $x=\log_b(M)$ and $y=\log_b(N).$

a) Use the definition of logs to rewrite $x=\log_b(M)$ as an equivalent exponential equation. Do the same for $y=\log_b(N)$.

b) Multiply together your results from part (a) and use properties of exponents to simplify.

c) Your result from part (b) should be a single exponential equation. Use the defintion of logs to rewrite that equation as an equivalent logarithmic equation.

d) Your result from part (c) should be a logarithmic equation that has an $x$ and a $y$ in it. Plug in $x=\log_b(M)$ and $y=\log_b(N)$ into that equation.

e) What can you say about $\log_b(MN)=$ ?

The Logarithm of a Power

$\log_b(M^p)=$ ?

To derive a formula we again make a substitution. Let $x=\log_b(M)$.

a) Use the definition of logs to rewrite $x=\log_b(M)$ as a equivalent exponential equation.

b) Your result from part (a) should be a single exponential equation. Raise both side of that equaiton to the $p^{th}$ power.

c) Use the definition of logs to rewrite your result from part (b) into an equivalent logarithmic equation.

d) Your result from part (c) should be a logarithmic equation that has an $x$ in it. Plug in $x=\log_b(M)$ into that equation.

e) What can you say about $\log_b(M^p)=$ ?

The Logarithm of a Quotient

$\log_b\left(\frac{M}{N}\right)=$ ?

a) Use the properties of exponents to rewrite $\log_b\left(\frac{M}{N}\right)$ as the logarithm of a product.

b) Use the product rule to split up your result from part (a).

c) Use the power rule to simplify your result from part (b).

d) What can you say about $\log_b\left(\frac{M}{N}\right)=$ ?

I still don’t care if the students know how to derive these properties. Rather, my goal here is to help them understand why these properties hold. In doing that I hope that they will also remember the properties better.

Check it out here.

The Cal Poly Math Department has a newsletter they publish annually called Polymath. I was fortunate enough to be featured in this years edition. My research partner and I wrote the article on minimal surfaces. You can check it out here.

1. Finish grad applications (by this week at the latest or I’m not going anywhere)
2. Study for algebra qual (now!)
3. Relax (or try to)

This list is in order of importance. One is the most important, four is the least.

Today is the 30th anniversary of John Lennon’s death. A truly inspirational human being, he is missed by many. The letter above was written by Yoko Ono in which she sent her blessings to the hundreds of thousands around the world who kept a silent vigil of affection and respect for her husband on Dec. 14, 1980. In celebration of his life, I encourage you to listen to your favorite Lennon songs and share them in the comments.

A few of my favorites:

• Imagine
• In My Life
• Love
• Watching the Wheels

P.S. November 29 was the 9th anniversary of George Harrison’s death. So be sure to listen to lots of George Harrison songs too!

I don’t watch much tv but I really like MythBusters. Science is about asking questions and making hypotheses and then testing them. And the MythBusters have so much fun doing that. For this week’s episode the MythBusters got an invitation to the White House and a personal request from President Obama himself. You can listen to a really cool interview from NPR’s Talk of the Nation here.

Richard Feynman on the rationale for scientific investigation.