The team that wins the coin toss is 21-25 all-time and the coin toss winner lost Super Bowls 35 to 41. All but three teams that have won the coin toss have elected to receive the opening kickoff. The Arizona Cardinals became the first team to defer to the second half in Super Bowl XLIII and lost to the Pittsburgh Steelers, 27-23. The Green Bay Packers deferred after winning the toss in last year’s Super Bowl en route to a 31-25 win and this year the Patriots deferred and lost to the Giants.

The coin flip result has had several streaks of 3 or 4 in a row. Streaks of 4 in a row are 24-27 4 heads; 32-35 4 tails; 37-40 4 tails; 43-46 4 heads. So if you want to bet on the coin toss next year I would say go with Tails since we have never had 5 in a row; but then again it is random.

Raw data from: docsports.com

I have used this number trick many times with beginning Algebra classes. I take a dictionary to class and before had write a word on a piece of paper then place it in an envelope.  At the beginning of class I select a student at random and give them the envelope then have each student pick a 3 digit number without any zeroes or repeating numbers, such as 389. Next have them perform the following calculations:

389 reverse the digits in the number to get 983.
Take the larger and subtract the smaller.
983 – 389 = 594
Reverse the digits of 594 and get 495.
Add them, 594 + 495 = 1089

Next I give the dictionary to a random student and have them take the result of their calculation (I check to ensure no arithmetic errors), use the first 3 numbers for the page and the last number for the word on the page.
Ask them to say the word aloud then have the first student open the envelope and say the word. The class is always amazed and soon realize they all got the same result in their calculation and wonder why – which is the segway to discussing some algebra.

The Proof

Call initial number abc = a*100 + b*10 + c
after reversing cda = c*100 b*10 + a
we will assume a>c (does not matter which one is larger).

A = abc – cba = (a – c)*100 + (b – b)*10 + (c – a)
since c<a we will need to borrow to do the subtraction (in our example above we had to take 3 – 9 for the units digits).
to borrow we take 1 from the 10s unit and 1 from 100s unit and get (since middle term of both numbers are the same we need to borrow from 10s and 100s units:
A = abc – cba = (a – c – 1)*100 + (b – b – 1 + 10)*10 + (c – a) + 10
A                   = (a – c – 1)*100 + (9)*10 + (c – a) + 10
now reverse A to get
B = (c – a + 10)*100 + 9*10 + (a – c- 1)

Next add them

A + B =  (a – c – 1)*100 + (9)*10 + (c – a) + 10 + (c – a + 10)*100 + 9*10 + (a – c- 1)
= (a – c – 1 + c – a + 10)*100 + (9 + 9)*10 + (c – a + 10) + (a – c – 1)
= 9*100 + (10 + 8)*10 + 9
= 9*100 + 100 + 8*10 + 9
= 10*100 + 8*10 + 9
= 1000 + 80 + 9
= 1089
same answer regardless of values of a,b,c keeping the all different and non-zero.

Also check this page for another set of calculations, Kaprekar constant, which will eventually end up with the same number.

It is produced by Dr. Singh and starts with Dr. Wiles talking about his excitement and  the fulfilment of a childhood dream; he is so taken as he talks about it he has to turn away to gain his composure. Moving for a math enthusiast or anyone else.

The video is well done and depicts well the struggles, effort and how he needed to bring together many existing mathematics in a very creative manner. Creativity is sometimes described as the ability to make unexpected connections. Dr. Wiles demonstrates this in many ways in his quest to prove the theorem and shows how his study of a branch of mathematics, elliptic curves, while at the time he thought it would not ever be of any help in proving the theorem (a period in his career when he had to put his interest in the theorem on hold) ended up being a major piece of his proof.

Take some time and enjoy, I think you will find it time well spent.

Isles: Have you ever tried to appreciate Euler’s number e? You know, the beautiful equation that connects three constants of mathematics? Have you?
Rizzoli: Yeah, I tried it once.

Unfortunately Dr. Isles mispronounced his name. She pronounced like the name looks but actually it is pronounced as ‘oiler’.

From Brown Sharpie’s mathematical comic site.

Euler's formula on spring break

“In this one-off documentary, David Malone looks at four brilliant mathematicians – Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing – whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God’s messenger and was eventually driven insane trying to prove his theories of infinity.”

Below are the two videos, very well done and well worth the time.
Part 1

Part 2

Drawing the Golden Rectangle

To answer your question would like to first start with your question about how to draw/create a Golden Rectangle.

To create a rectangle with this golden ratio:
1. Draw a square.
2. Extend two parallel sides.
3. Draw a line from the midpoint of one side of the square to the opposite corner.
4. Using the line you just drew as a radius, draw an arc between the two parallel lines.
5. Draw a perpendicular line from between the parallel lines from the intersection of the arc on the bottom line.
6. You now have a golden rectangle.
If the side of the Golden rectangle is A (thus we started with a square A by A) and the length is B then

This fraction, , is called the golden ratio (or golden section or golden mean).

The ratio is which is about 1.618304…
The ratio is so important it has its own special Greek character (Phi).
If you have ever heard of Fibonacci numbers, reference in the book and movie DaVinci Code, you may know they are directly related to the Golden Ratio. Fibonacci numbers are defined as the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … where the next term is the sum of the previous two terms. If you take the ratio of two sequential terms continually as you get further out in the Fibonacci numbers the ratio approaches Phi.

Golden Spiral and Fibonacci Numbers

Once you have drawn your Golden Rectangle you can continue on to create more Golden Rectangle and create a very interesting spiral – sometimes called the Fibonacci Spiral.

In geometry, a golden/Fibonacci spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of j for every quarter turn it makes.

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral which is sometimes known as the golden spiral.

Image Source: http://mathworld.wolfram.com/GoldenRatio.html

How Many on an 8.5 x 11 sheet of paper?

Now to address your question about how many Golden Rectangles can you put on an 8.5 x 11 piece of paper.
When you look at the construction of a Golden Rectangle you can see you can start with any size square. So the answer to your question is technically an very large number depending upon the size of the initial square.
If we look at the opposite of your question, what is the largest Golden Rectangle you can draw on a standard piece of paper. Using the construction method we need to start with a square, take half the side of the square and extend the side of the square to get the end point of the rectangle. We extend the square by:
Let say the side of our square is S. The length of the diagonal from the mid point of the side of the square to the opposite corner by the Pythagorean Theorem is:
. Thus the length of our rectangle is half the side of the square plus the length of the diagonal: which is = 1.618034…

For standard paper the length is 11 so we set the previous equation equal to 11 to find value for S to us as much of paper as possible.

Solving this equation for S we get

So we the largest Golden Rectangle on an 8.5 x 11 sheet of paper is width of about 6.8 by 11 inches.

Another way to look at answering your question about how many Golden Rectangles we can fit on a standard sheet of paper is to assume we are using a ‘unit’ Golden Rectangle which means the rectangle would have dimensions of side 1” and length which is about 1.618304…
If our side is 1 then would could have 8 rectangles high if we look at the paper in landscape mode. This will cover 8” of the paper from top to bottom. To see how many rectangles will fit across the paper we divide 11” by : 11/1.618034 = 6.798373828 so we could fit 6 rectangles across the paper. This gives us a total 6 * 8 = 48 rectangles to cover the paper.
If you use the same logic placing the rectangles with the paper in portrait mode you can fit 11 from top to bottom and 8.5 / 1.6.8034 = 5.2532888678 which rounds down to 5 so we can fit 55 rectangles in the portrait mode – a larger number than landscape.

There are many fascinating characteristics of the Golden Rectangle and would suggest you look at:
http://goldennumber.net/
http://www.mathsisfun.com/numbers/golden-ratio.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/

Dreams

In the broad daylight of mathematicians they check their equations and proof, leaving no stone unturned in their search for rigor. But at night under the full moon they dream; they float among the stars and wonder at the miracle of the heavens., they are inspired. Without dreams there is no art, no mathematics to life.

He is a British-Lebanese mathematician (born 1929), and widely considered one of the greatest geometers of the 20th century and 2004 Abel prize winner. Read his Wikipedia page for more details.

While looking at the web for more information on him I found this additional quotation from him on the Go Geometry site:

Visual Learning

Our brains have been constructed in such a way that they are extremely concerned with vision. Vision, I understand from friends who work in neurophysiology, uses up something like 80 or 90 percent of the cortex of the brain…
Understanding, and making sense of, the world that we see is a very important part of our evolution.

Therefore spatial intuition or spatial perception is an enormously powerful tool and that is why geometry is actually such a powerful part of mathematics – not only for things that are obviously geometrical, but even for things that are not. We try to put them into geometrical form because that enables us to use our intuition. Our intuition is our most powerful tool…

I think it is very fundamental that the human mind has evolved with this enormous capacity to absorb a vast amount of information, by instantaneous visual action, and mathematics takes that and perfects it.

Possibly his most famous quote is this one – keep in mind his passion for geometry:

Closing thought

Algebra is the offer made by the devil to the mathematician…All you need to do, is give me your soul: give up geometry –Michael Atiyah

You Do The Math: Explaining Basic Concepts Behind Math Problems Improves Children’s Learning

ScienceDaily (Apr. 10, 2009) New research from Vanderbilt University has found students benefit more from being taught the concepts behind math problems rather than the exact procedures to solve the problems. The findings offer teachers new insights on how best to shape math instruction to have the greatest impact on student learning.

The research by Bethany Rittle-Johnson, assistant professor of psychology and human development at Vanderbilt University’s Peabody College and Percival Mathews, a Peabody doctoral candidate, is in press at the Journal of Experimental Child Psychology.

Teaching children the basic concept behind math problems was more useful than teaching children a procedure for solving the problems “ these children gave better explanations and learned more”, Rittle-Johnson said. “This adds to a growing body of research illustrating the importance of teaching children concepts as well as having them practice solving problems.”

In math class, teachers typically demonstrate a procedure for solving a problem and then have children practice solving related problems, often with minimal explanation for why things work.

With conceptual instruction, teachers explain a problems underlying structure. That type of instruction enables kids to solve the problems without having been taught specific procedures and also to understand more about how problems work. Matthews said. “When you just show them how to do the problem they can solve it, but not necessarily understand what it is about. With conceptual instruction, they are able to come up with the procedure on their own.”

The study also examined whether having the students explain their solution to problems helped improve their learning. To test this, the researchers used the conceptual teaching approach with all students, and had one group explain their solution while the other did not. They found no discernible difference in performance between the two groups. While self explanation has been found to be beneficial in previous studies, Rittle-Johnson and Matthews found that when the students were given a limited time to solve the problem, the benefit disappeared. This led them to suggest that part of the benefit of self explanation may come from the extra time a student spends thinking about that particular problem.

Self explanation took more time, which left less time for practice solving the problems. Matthews said. “When time is unlimited, self-explanation gives students more time to repair faulty mental models. We found conceptual explanation may do the same thing and make self-explanation less useful.”

Rittle-Johnson is an investigator in the Vanderbilt Kennedy Center for Research on Human Development and in the Vanderbilt Learning Sciences Institute. The research was funded by the U.S. Department of Education.

Vanderbilt University (2009, April 10). You Do The Math: Explaining Basic Concepts Behind
Math Problems Improves Children’s Learning. ScienceDaily. Retrieved April 10, 2009, from http://www.sciencedaily.com­ /releases/2009/04/090410143809.htm

Teaching ideas from Dr. Wildberger:

In my opinion here are the keys to successful mathematics teaching at the university level, in order of importance. These might be useful to young lecturers who are starting out on their teaching careers.

• Content
  The fundamental requirement for succesful teaching (actually at any level) is to have something to teach. In mathematics, this means content that is accessible, useful and interesting. Material should be aimed at the appropriate level for students, it should be seen to be useful by them, and it should stimulate them at the same time. All three aspects are necessary, and getting the balance right is not easy. Well chosen examples are particularly important for mathematics teaching.

• Preparation
  The second most important factor for successful mathematics teaching is careful preparation. You must know the material, have your examples worked out beforehand, and have thought about how best to structure the content, both at the global and local levels.

• Delivery
  The third most important factor is effective delivery. Without distorting your natural self, you should strive to be enthusiastic, friendly, and to talk and write clearly. There are many specific skills to be learned in this direction.

• Action
  Mathematics is an active subject. Students learn not only by reading and listening, but also by writing and more crucially by doing. It is thus valuable to structure mathematics learning so as to provide opportunities for students to take lecture notes, work actively in solving problems, and to write up and discuss solutions.

• Resources
  The right tools make any job easier. A well-designed book is an important aid in succesful long term learning, particularly if written by an expert who has thought deeply about how to best structure the subject. Be skeptical about the merit of quickly put together notes, especially common on the web.