Algebra II/Trigonometry: Applications of Parabolas/Quadratics: Due Monday, November 16, 2015

Many of you have seen or played with the whispering discs on the east side of the science building.  How many of you know how they work?  Their unique parabolic shape helps to focus sound.  The following 4 min video discusses how a parabola directs sound.  Please watch this lovely illustration and explanation.

While we've studied quadratics a lot, we never seem to do enough applications.  In this blog, you will search the internet for applications.   Find at least one. If you discuss one, discuss in detail and well. If you discuss two, then you can write less about each.  Here are some hints:

So your job is to not just say, hey, it's a projectile path.... or hey, a parabola is the shape of those whispering things... your job is to explain why this is a parabola.  Find something about the FOCUS, or the LATUS RECTUM or the DIRECTRIX.  Explain something NEW about the parabola. Below is a hint. Ask Purple Math or Khan Academy to help you out with this.  I found HOT MATH to be the simplest explanation. 

Beware of catenary arches posing as parabolic arches. When you google "parabola" you'll find images of catenary arches. Be sure you are familiar with what those arches are before you write about them in your blog.  OR..... Write a blog about catenary arches.  What are they?  How are they formed?  They are kind of cool.  

Also beware of semicurcular Roman Arches.  These are not parabolic either.  In short, be careful about your choices. While you can use a parabolic model to fit catenary and semicircular shapes, this blog asks you to be more accurate mathematically and theoretically.  There's some bad information out there on line! 

Someone who understands the difference between catenary arches in architecture and parabolic arches in architecture is Ivars Peterson in his blog, The Math Tourist.  

Algebra II/Trig. NYT: The Importance of Recreational Math

You've heard me say, "Math used to be cool," or "It used to be cool to study math."  Some people still believe that doing math is cool -- some people don't recognize that some of the cool stuff they are doing is, indeed, math. And, well, you know, math is still cool.  Sometimes, perhaps we just make it into a process that we've come to call "math," but it's not really math.

Read this article from the New York Times.

Follow one (or more) of the links in the article, or simply google "Vihart."

Here are a couple links to Vihart pieces you might find entertaining:

Hexaflexagons. 

Elephants. 

I hope this is fun for you; please keep asking yourself the "Why?" or "How?" questions so you can delve a little more deeply into your blog.  Find something fun or interesting and write about it.  Let is be recreational!

Everyone: Symmetry, Palindromes and the year 2015. Due April 15.

Symmetry is a topic we look at in many areas of mathematics.  We've seen even functions, odd functions, and relations symmetric to the x axis. Because of the elegance, simplicity and beauty of symmetry, we are drawn to symmetric objects and ideas.  

 Palindromes represent a kind of symmetry; they read the same way forwards as backwards.  When written in binary form, 2015 is a palindrome: 11111011111. Even better, when factored completely, 2015 can be written as a palindromic number: 13*5*31. This won't happen again for another 30 years.

Some other non-numeric palindromes: 

race car;  

A man, a plan, a canal, Panama; 

Madam, I'm Adam.

And my all-time favorite:

Go hang a salami I'm a lasagna hog. 

Your task for this (hopefully) fun blog, is to find TWO new palindromes that are interesting. One MUST be a numeric palindrome with some interesting characteristic that is not just the symmetry, the other can be any kind (a word or a phrase or a number with an interesting characteristic).  Feel free to search the internet, but credit your sources.  Feel free to make up your own.  Explain the symmetry, say why it's interesting to you.  Describe the characteristics of your palindromic number.  

3-D Polar Graphing. Q4 Blog 1 Due March 20, 2015

You saw the basics of two dimensional polar graphing.  Che asked if there was a way to graph in 3-d in polar form.  There's a couple ways that you'll consider in Calc 3 in college, and there's a way you can think about it now.  Some of you actually use this all the time.

First, in Calc 3, you'll be using integrals to find cylindrical coordinates or spherical coordinates.  Cool stuff that you can see down the road.  Have a look at the links.

But here's an idea for you that's a commonly used and known form of three dimensional polar graphing. 

So look at this interesting form of three-dimensional graphing that we all have some familiarity with. After you look at it and consider it, then look at the task below (The following questions/ideas won't make sense if you haven't yet followed the link above headed "three dimensional polar graphing")

Your task in this blog:

1. Find the three-dimensional polar coordinates of some places that you've either visited or would like to visit.  Find the coordinates of "home."

2. There are some lines called "meridians" -- or some lines that have some important significance.  What are these and how are they used?  Have you ever crossed any?  What happened (if anything....)

3.  Your choice.  Find something interesting relating to this idea (and math) that you'd like to read and write about.  Have fun.

Jobs that Didn't Exist 10 years ago and GOOGLE.

So, one of the ideas I  keep bringing up in class is that the jobs that will be available to you -- jobs you'll hopefully love, find inspiring, lucrative, satisfying, and a big part of your life -- don't currently exist.  Indeed, many of the jobs held by young Americans NOW didn't exist when they were in high school or even college. Some examples: analyst of marketing data (did people purchase from your company using an iphone, computer, or store for example?); marijuana packager in Colorado; APP developer; sustainability director.)

So how do you prepare for these jobs that don't exist?  How did other folks who now hold these brand-new job opportunities develop the skills they use in their current jobs?  Great questions.  I'm convinced you just need to develop good, clear thinking skills, organizational strategies, and good habits of mind.  You can start working on those now; they are universal skills.

So, for this cycle's blog, there's two options.

The first: think about these "new" jobs that include math skills. Look around, or GOOGLE something to see if you can find a job that definitely did not exist 5 or 10 years ago. Please don't repeat those ideas shared already in class -- come up with your own NEW idea.  Describe that job, the skills necessary, and hypothesize how folks developed those skills to be able to do those jobs.

The second: read this powerpoint written by some of those great brilliant and creative geniuses from GOOGLE itself.  Find some cool, novel way they used math in their GOOGLE product.  There's lots of links provided in the presentation. You might not know what linear algebra is, but feel free to surf to learn something about different areas of math, such as linear algebra.  Find something appealing to you -- surely you can find something that rocks your boat in this presentation.  (How about that phrase, "rocks your boat" ?  It's a combo of "Rocks your world," and "Floats your boat."  Somehow, it seems to work better for me than both of the two more traditional phrases.)

Be sure your blog on this topic is substantive.  Write more than one 4 sentence paragraph. Develop an idea.  Be thoughtful and meaningful. You'll begin to lose points on your blog grade if you're  too brief or flippant or write too much fluff.  Make this fun and interesting for yourself and choose something that you think is cool and that you can expand upon. There are lots of options.

Probability in the Pool. Due March 4, 2015

I like to swim. When I'm visiting my dad in Connecticut, I swim mostly alone in a pool that is  50 meters by 25 meters.  Sometimes the lane lines were set to allow swimmers to swim "short course" (25 meters) or "long course" (50 meters).

On one of the short course days, I was alone in the pool.  There were 16 empty lanes.  I chose lane 8, one of the middle lanes.  I swam down-and-back (50 meters) in about 45 seconds.  This became a deliciously long swim as I moved back and forth alone in this pool with my own personal lifeguard.  Then, suddenly, I was jerked out of my fraction-calculating delirium (yeah, I practice fractions when I swim long distances) when waves overtook me.  A corpulent gentleman dropped himself and his furry belly into the lane next to mine at precisely the moment I was at his end of the pool.

Ah, I thought.  A probability problem.  (Notice I'm still in the realm of fractions!) What is the probability that he would (a) choose the lane next to mine and at the same time (b) choose to enter the water during the roughly 7 seconds that I am vulnerable to the tsunami he created at the near end of the pool?  And is this probability small enough that I should think this individual inconsiderate?

What if the lanes were set to be "long course" and the furry, corpulent gentleman deposited himself next to me during the 7 seconds I was at *that* end of the pool?  There are 9 lanes in this case; I'm in the middle lane.  It now takes 90 seconds for me to travel down-and-back. What's the probability that, if his entry is random, he would create his tsunami when it would disturb my swimming?

Well, now, TPC students, here's  your task. You can either choose to address the probability problems posed in this blog or you can choose to create your own solvable probability problems that you see in your daily life.

Have fun, as always.  Be sure your work is original (don't just copy someone else's work) and includes some significant thought. I encourage you to write something meaningful or real to you.   Feel free to be poetic or funny.

TPC and Misspelled: Fractions, Fractions: Due February 17

Quick -- which is bigger: 1/3 or 1/4?   Apparently, there is a significantly large proportion of the American Public that believes that 1/3 is less than 1/4.  After all, 3 is smaller than 4.

Evidence is cited in the New York Times Magazine, July 27, 2014 in the article (New Math)-(New Teaching)= Failure, by Elizabeth Green.  The premise of the article is that "the common core is the best way to teach math, but no one has shown the teachers how to teach it."  The example above appears in the article.

Follow this link; it has the same article but with a different title that's a little more provocative.  Read enough of the article to find the reference to the McDonald's restaurant competitor that offered the "third" pounder to be in competition with McDonald's "quarter" pounder. In your response to this blog, be sure to mention the more provocative title AND the name of the competitor restaurant, just so I know you read at least a little of the article.

I'd like your response to anything in this article in a meaningful but brief essay. Some ideas follow:  (Go ahead, be provocative!)

Do you think it's the education system? Do you think it's the poor education of our math teachers in this country? Many of your teachers (like me) at independent schools (like AA) have never received any formal training in math teaching. What's up with that?

Why do you think we have so much "innumeracy" in our fellow Americans (to use a vocabulary word coined by John Allen Paulos).  If we had as much "illiteracy" as "innumeracy," then people would be up-in-arms.  Why are folks not so upset about the "innumeracy"?  What's up with that?

Do you have examples of innumeracy that you'd like to share?  I wrote a post on this last year; I've made it available again if you'd like to see that.  (Find it below....)

Due Date: The THIRD day 9 in our semester 2 occurs Feb 17.  If you do not have class that day, of course it's due Feb 18.

To Infinity and Beyond. Due Feb 2

Infinity.  Several things we said in class:

Consider a segment of length 1 mm.  Then consider a line of infinite length in both directions.  There is a one-to-one correspondence between each point on the segment and each point on the line.  Yep, there's the same number of points on the segment (infinite) as there is on the line (infinite).

Let's say you are the proprietor of a hotel with an infinite number of rooms.  You have the good fortune of having all rooms rented.  Yet when a new customer walks in the door and asks about lodging for the night, you say, "Why yes, of course, I have space for you...."  There's a variety of ways that can happen, but the solution requires involvement with infinity.

 We can also use infinite sets in calculating probability.  The example given in class:  If a Natural Number is chosen randomly, what's the probability that the number will be a multiple of 8?  (1/8).

Do these examples blow your mind?  Find another story about infinity and write about it in your post.  Have fun; include jokes if you'd like.  But the jokes must have some kind of foundation or understanding or exploration of the infinite.

PreCalculus. BIG Numbers. Due Feb 2

So it's back in hunter-gatherer times for our species; around the campfire one night, there's a couple of cave people geeking out and talking about numbers.

Thor: "I can think of a number bigger than you can."

Lana: "Oh yeah?  Try me."

Thor: "Two."

Lana:  "Three."

Thor:  "Oh man.  Got me."

So how big is big?  We can talk in class about a billion seconds as being a lot of seconds -- nearly 32 years worth -- and say WOW that's awesome...what big numbers..... but what about the number of seconds that's in a billion years?  That number makes our billion seconds seem pretty trivial.  Just as we may chuckle at Lana and Thor, there may be a group of people several thousand years hence (assuming we don't kill ourselves before then) that laughs at our inability to contemplate what they consider to be large numbers.  This is almost like my 1986 computer I told you about where 40 megs was SO MUCH SPACE that people thought, "What is she ever going to do with that space?"

So how big is big? Exponential functions (with a base greater than one) are rapidly increasing functions and just beg the question: how big is that big number?  Choose your method of describing large numbers; give some kind of numerical idea of how we might contemplate large ideas.  Other than Jenny Lee's or Tony Borek's idea of how much time is equal to 1 billion seconds, what are other ways that might allow us to wrap our minds around large numbers? That's your task.  Yep, pretty open-ended and this asks you to think a little creatively and outside the box.  Google whatever you'd like, but be original and write your own stuff.

Let me give you some starter examples of what you can think about: How many blades of grass are there on AA property?  How about how many gallons of water are in Lake Superior? How many pennies would you stack together to reach the moon?

Have fun.  If you have questions, let me know.

Geo/Algebra 2, misspelled: Sets of Numbers Due Jan 15

Watch Sal Khan talk about sets of numbers.

Compare what he talks about to what we do/did in class about sets of numbers.  I hope that his lesson will help clarify the sets of numbers discussion from class. Know that you can always use his lectures/site for helping you learn the material from class. I know some of you already do.  If you don't know of this resource on line, then this blog is all about having you learn about Khan Academy.

In your blog, please write about how YOU will learn to keep these sets of numbers straight.  Find a mnemonic device that will help you remember, for example, the difference between the integers and rational numbers.  Go ahead, google to see if there's something that someone else has already come up with that you think will be helpful.  Or come up with your own idea.

Before you start on this blog, be sure that you make yourself familiar with the blog grading rubric.

Also, I'm thinking that we'll do a better job with blogs this semester; we'll have a blog due every day 9.  I'd like your responses, positive and negative, to this.

Precalculus: Welcome Back! Topic: Transcendental Functions. Due Jan 15

What does the word, "Transcendental" mean?  In English literature, you may (or may not) have studied the "transcendentalists" like Emerson or Thoreau.  Why are these authors called the Transcendentalists?

In mathematics, we discuss "Transcendental Functions."  Trigonometric functions are transcendental. We are now exploring exponential and logarithmic functions which are also transcendental.  Why?  How?  What does this have to do with the use of the word, "Transcendental" in English literature?

Before you write this blog, you'll want to read the instructions and rubric as posted on canvas. Be sure you are doing what you need to be doing to earn all the credit you need in this blog.

Also, in your blog, let me know how often you think it's reasonable to write a blog.  Is once each cycle good?  That's about once every 2 weeks.  I'm thinking that we could make the blog due every day 9, unless of course, there's a test that day. I'd like your response to that.