Saturday, November 22, 2014

What is a Radian?

Last week I spent some time reviewing what the CCSS say about trigonometric functions. As I was reading I came across a standard that made me pause.

CCSS.MATH.CONTENT.HSF.TFA.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

So I'll be honest, radians are still a bit mysterious to me. So I did some digging around in Sameer Shah's Virtual Filing Cabinet, and I came across Kate Nowak's blog post encouraging teachers to ask their students "what is one radian?" I then decided that I would write a post doing by best job to explain what is one radian.

Roger Cotes is credited for the concept of radian measure in 1714, however the term radian did not come about until 1873. The radian is considered the standard unit of angle measure. Radian measure depends on arc measure on the unit circle.

The measure, in radians, of an angle subtended by a circle arc is described as the length of the arc on the unit circle, divided by the radius. Wait...what is an "angle subtended by a circle arc"?

So this means that one radian is the angle subtended by an arc that is equal to the radius of the circle. So in other words an angle with measure 1 radian has an arc length of 1 on the unit circle.

An angle with radian measure pi (which we know is 180 degrees) has an arc length of 3.14159...Which means there are pi radians in 180 degrees.

So again, one radian is the angle subtended by an arc that is equal to the radius of of the unit circle. 

Another observation that comes from the use of radians is that the circumference of a circle is 2pi times the radius of the circle. The circumference can be thought of as the length of the arc of the entire circle, or as my high school math teacher taught me, the distance it takes to walk all the way around a circle.  So this would mean that the circumference of the unit circle is equal to the radian measure of a 360 degree angle ,  Well we know that the radian measure of a 360 degree angle is 2pi, so that means that the circumference of the unit circle is also 2pi. When we say that the circumference of a circle is 2pi*r, we are really just counting the number of "radii" that make up the exterior of the circle. 

In most mathematics angles are generally measured in radians because radians are more "natural", there are no degrees or meters attached to this measure so this allows radians to be unitless. It is very important that students understand the concept of radian measure, not just because the Common Core State Standards state its important but because using radians makes math easier! Trigonometric functions are defined by ratios in the same way radians are defined by ratios, so it only makes sense to use radians while working with trigonometric functions. However, it is important that students have a deep understanding of radians before they begin to work with them, it is important for students to know exactly what a radian is. When students have conceptual understanding of ideas like this it will only help their understanding of other mathematical concepts. 

Monday, November 3, 2014

Enrico Fermi and Walt Disney

So for those of you who don't know already, I love Disney. Those closest to me might even call me obsessed. I even have a Mickey Mouse tattoo and a tattoo with the coordinates to Cinderella's Castle in Magic Kingdom. Ok, so I will admit it....I am obsessed. Last weekend I took my 17th trip to Walt Disney World in Orlando to celebrate Halloween and enjoy Disney's Food and Wine Festival. While I was in Disney I of course discovered new rides and new shows that I had never even seen before, each time I go down I am amazed at how much of WDW I still have yet to experience. So this got me thinking, how many days would it take to do everything in WDW?

So in case you are not very familiar with Disney World, here are some facts for you:

  • On Disney property there are 4 amusement parks (Magic Kingdom, Animal Kingdom, Hollywood Studios, and Epcot), two water parks, a shopping district (Downtown Disney), and 34 different resorts, all which total 27,258 acres (42.59 square miles). 
  • 50,125,000 people visited WDW in 2013
  • In the 4 main amusement parks there are 71 different rides and shows

So now that I have decided to tackle my Disney Fermi problem, my next step was figuring out what "doing everything" means to me. I decided that this would mean riding every ride and watching every show (a show being some sort of entertainment that I would watch that was not a moving ride) in all four of the main parks. I decided to exclude watching parades and fireworks because those vary depending on the day and also the time of the year. I also excluded anytime spending shopping. So next I figured out what information I would need to solve this problem, and this is what I decided was important to know:
  • Average wait time for each attraction (I used a 10 minute wait for each show)
  • Ride length for every ride and the run time for each show
  • Average time spent walking from ride to ride
  • Time spent getting to the park (I used the Disney bus system because it is provided for free for people who stay at a Disney resort)
  • Time getting into the park and leaving the park
  • Time spent eating
  • Park hours for each of the four parks
Note: I decided that for all of my approximations I would round up my values to account for bathroom breaks, longer than average wait times,  crowds etc. 

So most all of this information I was able to find online. I even found a website that lists the exact times for each ride and show in WDW. Another cool website I found complied average wait times for every ride depending on the crowd. Information I was not able to find online I just estimated using my experience of my many trips to the parks. So once I collected all of this information I compiled it into one huge spreadsheet and then began to add everything up. 

Except it wasn't that easy. I decided that I would figure out how many days it would take to complete each park first and then add all of those days together. Some parks took longer than one day to complete so on those extra days I had to add extra time for eating and travel time. This was the formula I used to calculate the number of days each park would take.

# of Days = [Travel Time + Time Spent Entering Park + Meal Time + Total Average Wait Time + Total Ride Time + Time Spent Walking] / [Park Hours]

Then if the number of days was greater than one I added more travel time, time entering the park, and meal time accordingly. 

After all of my calculations I came up with these values for the number of days it would take to do everything in each park:
  • Magic Kingdom: 2.462 days
  • Hollywood Studios: 1.26 days
  • Epcot: 1.65 days
  • Animal Kingdom: 1.26 days
 This means that it would take roughly 6.63 days to ride every ride and watch every show that WDW offers. Based on my experience I figured that my answer would be close to a week once I figured in the park hours, travel time, etc. Then I began to wonder how much it would cost to take this trip to Disney to do everything. So then I began another hunt for more information. I decided I would estimate this seven day trip using a family of four (I figured this was a common family size). 

These were the expenses I found:
  • 4 round trip plane tickets from Grand Rapids to Orlando: $1,836.80
  • 8 night stay at Disney's Caribbean Beach Resort: $1,781.68
  • 7 days worth of admission to WDW: $1,358.94
  • Cost of food (I used the Disney Dinning Plan estimate): $464.16
So to take your family of four to ride every ride and see every show in WDW it would cost roughly $5,441.58. And this doesn't even include any shopping. So after spending time working on this problem I decided to make myself a Disney checklist, and it is my current goal to finish this checklist. I guess that just means I have an excuse to visit my favorite mouse another time!