Pascal's Triangle Response

Pascal's Triangle

One of the coolest and most useful pieces of mathematics that you learn in high school math is what we call Pascal's Triangle.  I've copied a few rows of the triangle below:












 If you don't know how to form the values in the hexagons of this triangular shape, you'll want to look that up and explain it in your blog. 


A couple of the interesting things include the following.

Find the sums of each row.  Then compare these numbers.  What pattern is represented?

The powers of 11 are represented in this triangle; how so?  Use the triangle to find the seventh power of 11.  Explain how you've done this.  

Triangular numbers, square numbers, Fibbonacci numbers, prime numbers.....all appear in interesting patterns in Pascal's triangle. This interesting site will give you a simple introduction.

There's also some amazing visual patterns that appear in Pascal's triangle.  Explore those and report back what you find.  (I know there are some artsy folks out there -- this part's for you.) 

In your blog this time, write something about Pascal's triangle.  You can be expansive.  Perhaps you'd like to write about how the Chinese people knew and used this triangle long before Pascal was born.  Perhaps you'd like to write about Blaise Pascal himself: when did he live and what influences did he have? Of course it would also be wonderful for you to find something new and cool about the number patterns Pascal's Triangle. 

Have fun.  And. If you find some good piece of math humor, feel free to include that, too. 

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To get each value in the hexagons you must add the two above. For example the 4 in Row 4 comes from the one and three added together. 

http://www.mathsisfun.com/pascals-triangle.html

Row                                                                 Sum of Values in Hexagons

Row 1	1
Row 2	2
Row 3	4
Row 4	8
Row 5	32
Row 6	64
Row 7	128
Row 8	256
Row 9	512
Row 10	1024
Row 11	2048
Row 12	4096
Row 13	8192

The values of the hexagons in each row is equal to 2 to the power of the number of the row.

The powers of 11 are also shown, you just have to read it diagonally. But when there is multiple digits you would sometimes have to carry a digit over. For example the 7th row shows the numbers 1, 7, 21, 35, 35, 35, 21, 7, 1. 11 to the 7th power is 19487171. The 7 and 2 in twenty one combined to make nine, The 1 in twenty one combined with the 3 in thirty five to get four, and so on. 

 A new pattern I have discovered in Pascal's Triangle is that a number squared is equal to the number to right and the one below it added together.,

http://www.mathsisfun.com/pascals-triangle.html

Pascal was not the first to discover this triangle. A Chinese mathematician, Chia Hsein, in the 11th century used this to “was using the triangle to extract square and cube roots of numbers” (Clawson 133). http://pages.csam.montclair.edu/~kazimir/history.html

http://www.ualr.edu/lasmoller/pascalstriangle.html

http://www.quickmeme.com/Math-Major-Sloth/page/10/

Math Humor

http://www.funnyjunk.com/funny_pictures/1074272/Math/


http://www.pinterest.com/pin/85216617920386605/


http://www.damnthatswhack.tumblr.com 

http://www.reichenbachhhero.tumblr.com

http://www.anexperimentallife.tumblr.com

Fishing For Complements

resilientknee.tumblr.com

http://www.785dudeman.tumblr.com

http://www.binichhier.tumblr.com

http://www.joshfuckingwebb.tumblr.com

Classy math joke

What is a geometry student’s favorite place?

The beach, it is full of tan gents.

http://goodriddlesnow.tumblr.com

TPC and GA2: Human Calculator Skills Response

TPC and GA2: Human Calculator Skills

 

Yesterday in class, Elijah Ray made a swift calculation in his head.  Occasionally, someone in class blurts out the product of a couple of two-digit numbers with lovely speed.

When can YOU do this?  Under what circumstances can you multiply swiftly?  Do you ever make a (let us all pause here) geometric illustration of your calculation in your head?  Can you picture something with, say, algebra tiles?  Let me illustrate.

 Let the following illustration of a rectangle represent the product of 43 and 42. Let the horizontal represent 40+3 and the vertical represent 40+2.   The blue square represents 40x40; the green bars each have dimension 1 x 40; the yellow squares each represent a 1 x 1 unit. 

 

 One (of three) options for your blog this time is to relate this idea to areas.  Demonstrate how these tiles support or illustrate the traditional way to multiply these two numbers together:

 

 

...or how the tiles support FOILing:




But there's more. I'd like to draw your attention to two modern-day mathematical geniuses  from India. 

One woman, Shakuntala Devi is known as "The Human Calculator" for her ability to calculate products of large numbers in her head very swiftly.  From the on-line version of the India Times:
Shakuntala Devi once competed against a computer to see who could come up with the cube root of a 9 digit number first and she defeated the computer at this challenge. The same year, in 1977, Shakuntala Devi was asked to give the 23rd root of a 201-digit number; she answered in 50 seconds! On June 18, 1980, she demonstrated the multiplication of two 13-digit numbers 7,686,369,774,870 × 2,465,099,745,779 picked at random by the Computer Department of Imperial College. The correct answer was presented by her in just 28 seconds!

I've always loved stories of Srinivasa Ramanujan. One of my favorite stories is the following, fromDurango Bill's site:
    If you mention the number “1729” or the phrase “Taxicab Problem” to any mathematician, it will immediately bring up the subject of the self-taught Indian mathematical genius Srinivasa Ramanujan. When Ramanujan was dying of tuberculosis in a hospital, G. H. Hardy would frequently visit him. It was on one of these visits that the following occurred according to C. P. Snow.

   “Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’”

A second option for your blog this time is to search the web for yet another "human calculator." Have your mathematician be from relatively modern times (the last 100 years) and write something about him or her.  Include your internet sources in your blog.

A third option for your blog this time is to find an interesting story about one of these two folks that I identified above that are NOT on the sites I've provided for you.  Also be sure to include your internet sources in your blog.

                                                                                                                                                                  

Scott Flansburg hold the Guinness World Record for the "Human Calculator". Flansburg has been teaching for more than 20 years. He is able to subtract, add, multiply, take the square and cube roots of numbers in a short amount of time. He said the only that holds him back is that he cannot speak as fast as he gets the answer. I think it is really interesting that he can do math in his head faster than professionals on calculators. He has appeared on many talk shows such as Oprah, The Ellen DeGeneres Show and ABC’s Good Morning America.

The Human Calculator reminds me of when I was watching Live! with Kelly and Michael when a teen came onto the show. He was wearing a math Olympiad shirt and he solved mathematical questions against Kelly and Michael. he answers the questions way faster than than the two hosts. The questions aren't as complicated and complex as the ones Scott Flansburg does, but he is still impressive. I am unable to find a link or video on him. I just remember sitting on the couch with my mom trying to see how fast I could solve the problems.

The Human Calculator also reminds me of when I was in about 4th grade we did a race against another student on a calculator. It was simple multiplication to 12 x 12. Almost all the students were faster than the calculator. Once you commit the math to memory it becomes easier and faster to yourself.

Sources:
http://scottflansburg.com/