Palindromes

I think that palindromes are very cool! This was a neat idea for a blog and a fun thing to explore.
There are so many different palindromes that people have discovered.  I'm still in the process of trying to think of my own.  The most fascinating non-math palindromes are the ones that aren't just one word, but whole sentences, paragraphs even.  It's crazy what people can come up with (and maybe how much time they have on their hands (; )!
P.S During my research, I found that palindromes are used in a lot of computer codes!

Some of the palindromic square numbers are: 0, 1, 4, 9, 121, 484, 676, 10201, 12321

If a number n, is followed by any number of zeros, and then the number n again, it will be a palindrome. i.e 10000001 or 2002 or 3000000000000000000000000000000000000000003.

Goat or Car Problem Confusion

I have spent a long while thinking about the goat or car problem, and I am totally perplexed.  I have gone over the tree diagrams over and over, and I just do not understand how switching your door choice to the other door would do anything for your chances at all.  Of course, we know, because we are looking at what is behind each door, which door should be chosen, but the person choosing the door doesn't know that, so I have no idea why your chances would be increased.  I've spent a lot of time thinking about this.

I don't see any logic involved in the tree diagram.  I am probably just confused, but this problem is totally and completely perplexing to me.  I decided just to write about my confusion because it was bugging me so much.  I would love for this to be better explained to me though.  I spent about 45 minutes looking at the tree diagram and trying to figure it out.  Everytime I thought I understood why the chance of winning a car would increase to 2/3 rather than 1/3 when the game show contestant changed their choice, I realized I was still just confused!

I would say that this was a successful blog prompt because it made me think. It challenged me and frustrated me, and challenging is what math should be.  I will continue to try and understand this problem.

The Probability of Missing a Gate

I thought about trying to solve the swimming pool problem, but I also wanted to relate probability to my daily (or almost daily) life.  I started thinking about the nerves that build up before a ski race.  You have to finish the race, and make it around every gate, so all of the racers ask themselves this question: What if I miss a gate?
I then started thinking, what is the probability that I will actually miss a gate in a given race?

One particular course that I have raced on has 40 total gates, alternating red and blue.  On any given gate, there are two options: I can either make the gate, or miss it.  So, the probability of me missing a gate is (1/2)(to the 40th power).  (*I don't know how to do the formatting on here).  So really, I shouldn't be that worried about missing a gate.

It is interesting to me how skewed our perception of probability is.  We think we know, and we worry about things, but the numbers show us that the reality really isn't that scary!

Fractions AKA American Troublemakers

As I read Ms. Mariner's blog about fractions, and Americans not knowing the correct sizes of fractions, I was not surprised.  Although I cannot recall a specific time when I have encountered this misconception of fractions, I totally believe that it is a real issue.  Sometimes we Americans, admittedly myself as well, are just plain dumb, but who's to blame? Math teachers? Students?

The Common Core is quite frankly something that annoys me, a lot.  Ms. Mariner commented on how her and other independent school math teachers don't go through any sort of training.  I think that this is WAY better than them going through training.  If we wanted all of our math teachers to be exactly the same, we would be taught by robots, but robots would not be effective!!! By them not going through training, they are able to teach math in a way that makes sense to each individual student.  I think that's our country's education problem.  We want everyone to be the same! Well America, BUMMER DEAL.  We're not all the same, and our learning needs are not either, so suck it up and teach our students the way they need to be taught.

As far as the A&W/McDonald's issue mentioned in "Why Do Americans Stink at Math?" I think that it is a direct result of American's not understanding what they're learning.  Sure, they memorized that 4 is bigger than 3 when they learned how to count to 10 in kindergarten, but did they ever understand what fractions meant?  It was all memorization, so when they went to apply it in the real world, making purchasing decisions, they didn't get it.  Pure memorization of simple concepts allows absolutely no room for understanding and application of slightly more complex concepts.

This is just a little bit of evidence for why Common Core won't work.  We are not robots, our teachers are not robots, and we should not act as if we are.  We are individuals, with different skills, and different ways of discovering things.  We must cater to that if we truly want to understand and apply learned skills to our lives.

BIG Numbers

It is easy to be overwhelmed when we think of big numbers.  Not big numbers like 100, but REALLY big numbers. When I think about how to describe this overwhelming sensation, my mind goes to little kids.  It reminds me of when you're little, and you stretch your arms out as wide as you possibly can, and your left hand is as far as humanly possible.  You look up at your mom or dad and say "I love you THIIIS much!!"  and your parent looks back at you and says "I love you to the moon and back!"

...Do we ever think about just how much that is? I don't think we do, and I would argue that as we get older, we really don't understand huge distances any better than we did when we were 5, 6, or 7.

When a little kid stretches their arms as far as they can, that distance between their arms is their "forever" distance.  That is their standard for "the most in the whole wide world."  When we tell people that we love them to the moon and back, that is our "forever" distance.  Yes, we logically know that there exist distances larger than that, but we don't consciously comprehend that.  A little kid knows that there exists a distance larger than the space between their palms, but they don't consciously comprehend it.

This is all due to standards.  We created standards in our minds, and that is what we compare everything we discover after the standard is set to.  To test my theory, I asked a 7 year old girl, Maci, what the biggest number she knows is.  She said that 100,000 is the biggest number she knows.  100,000 is Maci's standard.  She cannot comprehend numbers bigger than that, because they are too big when compared to her standard.  When she thinks of small numbers, she compares them to her large standard of 100,000, and she is able to comprehend them.  As we get older, our standard gets larger, but when we hear big numbers we are still comparing them in the same way that 7 year old Maci does.
Everyone has different standards for their "forever" numbers, and that is why we individually comprehend different numbers in different ways.  An example of this is a runner.  Someone who runs 3 miles normally, cannot comprehend running 10.  This is because 3 is their standard.  Someone who runs 26 miles normally, can easily comprehend a 10 mile run.

Thus, it is all in what we are used to.  Standards are the key.

Transcendental Functions

I started by pulling up Webster's online dictionary and searching the word, "transcendental." I was directed to the definition of "transcendent". This is defined as "going beyond the limits of ordinary experience," or "far better or greater than what is usual."  Now, I don't think that reveals to me how the word is related to math, but I do think that I can think of it in such a way that it does.  For example, maybe we are studying transcendental functions because we want to take our minds to a place where they go beyond the limits of an ordinary mathematical experience, or because we want to study mathematics in a way that is far better or greater than what is usual.  These definitions and applications intrigued me.

Next, I found out what transcendentalism is.  It is a practice that I am not sure I am in agreement with, but that is interesting to ponder.  It is a practice of people who do not accept things as religious beliefs, but rather as a way of understanding life's relationships.  This also got my "math brain" thinking.  Maybe we are studying transcendental functions because we want to study the relationships between numbers and equations.  We want to discover how things work together in a mathematical context.

Finally, as I did research on transcendental functions themselves, I found my wonderings to be almost correct.  Transcendental functions have the name they do because they cannot be expressed simply.  They cannot be expressed using simple algebra.  They allow us, and force us, to go deeper into our mathematical thinking.

Sinusoidal Curves in Ski Racing

As I thought about this blog, and what I should write about, I just couldn't find much inspiration.  I began to think of ways that I could relate sine, cosine, or tangent to my life.  Since ski season is just getting started, I naturally thought of ski racing.  I believe that a sinusoidal curve can be used to model the turns that a ski racer makes around the gates on any given course.
Here's a picture to refer back to, in order to make what I say more clear:

As you can see, there are various different aspects of sine curves that are evident in a series of ski turns:  

Observation 1: The horizontal distance from red gate to a point collinear with the blue gate is the amplitude of the sine curve.
Observation 2: The vertical distance from the red gate a point collinear with the blue gate, is the wave length of the sine curve.
Observation 3: The distance from blue gate to blue gate, is the period of the sine curve.

I think that it is neat that I could apply this to my skiing.  While I was skiing this past weekend, I was thinking about these sine curves and what I was creating with my skis. Math really is everywhere!