An interesting pattern with derivatives

I found an interesting pattern while finding  higher order derivatives using the power rule. For readers who are not familiar with derivatives you might want  to watch the video here ( MIT open course ware derivatives.)

I was curious what the 3rd derivative of  x⁵ was.
So I used the power rule  to find the first derivative and I got f¹(x)= 5x⁴ I applied the rule again to get the second derivative and got f²(x)= 20x³ I once again used the power rule for the third derivative and found the third derivative was f³(x) = 60x² Now by this time I started to get curious as to what the Nth derivative,fⁿ(x) would be for functions using the power rule. So I began to to work it out algebraically. I started with the general function f(x)= x^a and began taking the derivative by continuously using the power rule, and got the following results:

1. f ¹(x)= a x ^a-1
2. f ²(x)= a (a-1) x ^a-2
3. f ³(x)= a (a-1)(a-2) x ^a-3
4. f ⁴(x)= a (a-1)(a-2)(a-3) x ^a-4
5. f ⁵(x)= a (a-1)(a-2)(a-3)(a-4) x ^a-5

From these a pattern begins to emerge with, a , as I took the higher order derivatives. Here is the pattern of  ,a, throughout each of the derivatives:

1. a
2. a (a-1)
3. a (a-1)(a-2)
4. a (a-1)(a-2)(a-3)
5. a (a-1)(a-2)(a-3)(a-4)

As we can see it looks like a pattern similar to a! or a factorial, but this alone is not the pattern we need to a be able to see how this relates to the nth derivative. Well lets look at the derivatives. The second derivative of  f(x)= x^a  is f²(x)= a (a-1) x ^a-2 ,and from this we see that it is not just a! it is a!/(a-2)! because a! keeps going until (a-some number) = 1  For the second derivative to make sure a! does not go past a (a-1) we use (a-2)! in the denominator because it cancels out all of the other factors past that point.  That brings us to the general form of a!/(a-n)!

Now with the a in the exponent  x^a we see that the a reduces by the the derivative we are on for example the second derivative ,  f²(x), the exponent is x ^a-2   That makes the general form for the exponent x ^a-n

To put all this together we get the formula for the nth derivative using the power rule as

f ⁿ(x)=  a!/(a-n)!  x ^a-n



 I am not sure if this pattern was already discovered by someone else or not. I have done some research and have not been able to find anything about it online.

Number Theory: Not as Scary as it Sounds

What is Number Theory?

                Number theory is simply the study of the properties of numbers. The topics in number theory range from a variety of fairly simple to extremely complex. (Most people can understand and follow along with topics in the middle of this range.) The simplest properties of numbers are still a topic of research in number theory today. A person can start becoming an amateur number theorist by simply picking a number and finding all of the ways that number can be mathematically manipulated. Then by paying attention to and recording any patterns that arise to see what new math tools you discover.  While doing this you can do a little research to find other ways to manipulate the numbers you chose, and through the induction/deduction process of logic find your own theories. You might not find anything groundbreaking, but you will certainly find fun ways to learn and enjoy math on your own. I found an interesting “shortcut for multiplying by the number nine by simply making a chart of all of the multiples of nine and analyzed it until I found a pattern. I am not sure if someone else has already found this or not, but it was a very fun “trick” to use. You can find a full explanation with examples here Multiplying by 9,99,999...etc

                In high school, you were taught some of the basic properties of multiplication.  Such as: the distributive property, and  the associative property. You were most likely shown a formula to explain these similar to
a(b+c) = ab+bc, Which describes the distributive property.

Well where did we get this?  It is simply a result of number theory, or another way to define it is the logic of numbers. It is difficult to tell who to actually give credit to for discovering this property; it is one of the most widely used properties in mathematics.  It was, however; discovered by analyzing the properties and patterns found when multiplying and dividing numbers. The use of the letters, or variables, comes from what is called abstraction. Abstraction is simply taking something out of its original context and making a general form of it. This is not a great definition, but hopefully you see that when we find a pattern that is very useful, we need to make a method to apply it to all numbers.

Now some may understand the distributive property better if it is shown with actual numbers. Like this 2(3+4) =  2x3 + 2x4  but without abstraction some people would look at the example using numbers and think it only applies to those numbers.  Effective abstraction leads into using proofs to check your pattern to see if it applies to all numbers or just a certain set of numbers of numbers with specific properties.  I will leave that discussion for another post. Hopefully, this will give you a basic understanding of what number theory is and inspire you to learn more. 

Create inspiration to learn math.

               One thing the current math education policies do not address is relieving math anxiety. Many educators and policy makers talk about it, but do they really have a solid plan for relieving it.  The common discussion is all about how math is taught, but what about the kind of math that is taught.  Some educators are already implementing styles of teaching that relates to what students are interested in. This is a step in the right direction, and this does increase an interest in math education.  Many educators and even individuals have taken to the internet to provide lessons, practice, discussion boards on math education and homework. When you look at all of these education sites they all mostly just reflect the same things we teach in schools.

 Does this inspire students to learn mathematics? Does this relieve math anxiety? One way to answer these questions is to look at the students that are actively participating in these. Are they doing better on tests? Maybe…  We might be improving scores in the small groups that are actively participating in these online programs, but once again we have the question, are we relieving math anxiety?  Maybe to a small degree..

A simple way to relieve math anxiety is to introduce students to simple math “tricks”, or other ways to simplify or shorten the math calculations that most people only know one way to do.  I have already presented a couple of these methods. Such as Long multiplication in Reverse also called left to right multiplication, or multiplication without carrying. The “left-to-right” method is a good trick which helps with mental math. Students can learn to answer more difficult multiplication problems in their head with relative ease.                               

   This will make them more confident in their math skills. It will also change the way their peers look at them. They will feel smarter and it will increase their desire to be smarter by learning more about math.  The “left-to-right” method is not the only “trick” to make multiplication easy. There is also Russian Multiplication also known as binary multiplication. This method is great because you don’t have to have the multiplication table memorized to use it. You just simply have to double numbers and cut numbers in half and then add.  

All of these simple methods make math easier to students, relieving their initial anxiety. With the many ways that we can use to make math easier, we can teach a new “trick” every day.  The whole point is to increase desire for learning and relieve math anxiety.  These are the two main obstacles in helping children not only have better understanding of math, but also to do better on tests.

Fibonacci Meets Pythagoras...

Here is an interesting pattern I found while substituting Fibonacci Numbers into the primitive solution for the Pythagorean problem.   The primitive solution to the Pythagorean problem is:

a² + b²= c²  
Given any two arbitrary integers  m  and   n 
 a =  n²  -  m²
 b =  2mn
 c =  m²  +  n²



I am not providing a full proof of this solution here. I am simply showing the solution because I use the solution to generate some very interesting patterns. I used Fibonacci numbers in the primitive solution and got the following results.... ↓ ↓ ↓ ↓ ↓ ↓ ↓


  Some of the Pythagorean triples from the chart are:
   3²  +  4² = 5²
  5²  + 12² = 13²    Look at the c terms do you see a pattern?.......                                                                  16² + 30² = 34²     That's right the c terms are all Fibonacci numbers.                                                          39² + 80² = 89²                                                                                       
                                                                                                                 
               If you look in the columns under the  m and the n you will notice that I have the Fibonacci sequence written in two ways the m column I started the Fibonacci sequence with 0 which still works with the pattern, and in the n column I have the Fibonacci sequence starting with 1.  I do this so when I substitute the numbers into the primitive Pythagorean solution They wont just zero out.


         There is also another interesting pattern I found with the Fibonacci/Pythagorean triples.  
If you look in the two columns on the right I have showed the place value the c terms are in the Fibonacci sequence. Example: the number 1 is the 2nd  number in the Fibonacci sequence and the number 2 is the 4th. And this is their place value when you start the Fibonacci sequence with 0.Using the column where I started the Fibonacci sequence  with 1 the corresponding c terms are in the odd place values instead.

To explain it another way here I will list out a few of the numbers from the c column

1. 1   is the 2nd Fibonacci number when the sequence starts with 0 and the 1st when it starts with 1
2. 2    is the 4th Fibonacci number when the sequence starts with 0 and the 3rd when it starts with 1
3. 5      is the 6th Fibonacci number when the sequence starts with 0 and the 5th when it starts with 1
4. 13    is the 8th Fibonacci number when the sequence starts with 0 and the 7th when it starts with 1
5. 34    is the 10th Fibonacci number when the sequence starts with 0 and the 9th when it starts with 1
6. 89  is the 12th Fibonacci number when the sequence starts with 0 and the 11th when it starts with 1
7. 233 is the 14th Fibonacci number when the sequence starts with 0 and the 15th when it starts with1 

I could probably explain this better,but in general most should be able to look at the chart and see the patterns.  There is also more patterns found here when doing this.  The one I like is the c column has a lot of prime numbers. a few of them are     2,    5,   13,  89,    233,    1,597,   28,657.
 I have carried this out as far as Excel will let me without throwing errors and all of the c terms are Fibonacci Numbers
I am curious to see what patterns the rest of you come up with.     Happy Hunting!!!!!!!!!!

Mathematics education in the technological age

      There has been a lot of buzz  about Silicon Valley’s role in mathematics education.  Computers and the Internet have provided a new outlet for people to share their knowledge.  You can find thousands of videos on nearly every subject on youtube.  Surprisingly, many of these videos are actually very informative.  There are many people out there with a significant amount of knowledge on the subjects they discuss in their videos.  Not all of them are educators; many of them are normal everyday people who simply have a passion and understanding for a particular subject.  Mathematics is a difficult subject for a large number of people, and that is why we see so many mathematics videos on youtube.  Included are many videos from the now controversial Khan academy.  The common argument we see is that the definition of mathematics changes as the medium it is placed in changes.

Here is a quote from Dan Meyer on his blog at  http://blog.mrmeyer.com/?p=12782

“YouTube videos, digital photos, MP3s, PDFs, blog posts, spoken words, and printed text are all different media and they are all suited for different messages. When you attempt to distribute mathematics through any of these media, it changes the definition of mathematics.”

To read the full post go here  http://blog.mrmeyer.com/?p=12782

            So basically the use of technology to explain mathematics has changed its meaning. I have two questions I am going to propose. 

First: Is redefining mathematics truly hurting people’s capacity, ability, or willingness to learn?

         

     To answer this first question let’s look at khan academy.  I just looked them up 5 minutes ago, and on their youtube channel they have 8.3 million channel views and 121,000,000 total upload views. Now according to the U.S. census bureau in 2010 there were approximately 83 million people age 3 and older in school.  This “implies” that, there are more views than the number of students in the U.S.  Now the total views do not represent the total number of unique views. It does however show a significantly increased interest to learn mathematics.

     Khan academy isn’t the only source of videos on mathematics circling the internet. It just seems to be the most “popular”.  Many individuals and other groups all post their knowledge of mathematics in different formats.

Second: If this were true for printed text, how did the printing of the very influential “liber abaci” in the 1800’s change its meaning from the original hand written manuscript?

       

  The use of the Arabic numerals 0-9 and place value are the universal standards today. Without the use of new technology and new mediums the Liber Abaci would have not have half the impact it did.

Did this in anyway hurt students’ ability to learn or understand mathematics? Or did this make it easier to understand and share this knowledge with the world.

      Now let us look at today’s technology. The computer and the internet have made a huge impact on the distribution of knowledge. (Just as the printing press did when it was invented.) The only difference is now with computers we can share our knowledge with the world nearly instantly. Mathematics has always been about analyzing and understanding the world around us. The sharing of knowledge has never changed that definition. It has only enhanced how many people are able to use mathematics for whatever they wish to analyze or understand.

           

    What role do educators, parents, and students have in education?

    

Degree holding, certified educators seem to have or desire to have a complete control over how students are educated.  I agree that educators who have spent years of their life studying their craft deserve a certain level of respect. However, we need to ask are they all still living up to the respect that their “title” deserves, or is the title “all” that accredits them. ( I am not saying that all teachers “hide” behind their title. I am just simply saying that it sadly does happen) Parents and teachers both are missing the most import factor that controls a students learning. That is the students themselves. Students learn from each other and from all of their experiences combined, more than they learn from schools.

    

     When you look at other countries that surpass the U.S. in math and science scores, most would wonder what they are doing differently.  In general U.S. students (k-12) are punished for discussing, disagreeing, or criticizing problems offered during class. (This is a generalization that does not “always” happen in “all “classrooms, but it unfortunately does happen)

In Japan, for example, we see a complete opposite practice.  The students there are encouraged to discuss and criticize the material in order to gain a full understanding of all ways to look at a problem.

To better explain this here is a quote from a case study from the National Institute on Student Achievement, Curriculum, and Assessment prepared by Angela Wu.

“ While students look to teachers for comprehension and evaluation in American classrooms, students look to each other in Japanese whole-class instruction classrooms. The teacher asks the class to evaluate individual students' solutions to math problems.”

The original report can be found here http://www2.ed.gov/pubs/ResearchToday/98-3038.html

    

    

 Group focused learning is teaching students to teach themselves, and each other. When we look at this we can wonder, how this relates to the flood of tutorial videos and other online media for teaching and learning.  The group learning mentality has evolved past the students in one classroom or even one country learning from each other. It has evolved into individuals who are not educators, but are more knowledgeable on a subject than their target audience. These individuals share their knowledge in the best way they can, to reach as many people who can benefit from it as possible.  The individuals that provide their knowledge for free should be given as much respect as any other educator whether they just teach the mechanics of a subject or teach the philosophy behind it.

   

 Do Silicon Valley and regular classrooms actually tell students the same thing?

Rote memorization and mechanical repetition of tasks is exactly how the “majority” of educators teach mathematics. It is a sad truth, but it is the truth. (I am sure a lot of educators would completely disagree with me on this, but even if I get 1000 emails/comments on this that number will still represent the minority of educators.)  

     A lot of the individual video tutorials found on the internet also provide this type of instruction. The big difference is that Silicon Valley or computer based instruction is available to everyone at any given time.

To quote Dan Meyer once again

“On the one hand, Silicon Valley tells students, "Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself." Our best classrooms tell students, "Math is something that requires the best of your senses and reasoning…”

 To read the full post go here  http://blog.mrmeyer.com/?p=12782

This statement implies that there is a separation between Silicon Valley and the “…Best classrooms…” Although this may be in part true, the number of the “best” classrooms is not a big number.  I do completely agree with Dan’s last statement    "Math is something that requires the best of your senses and reasoning…”

Math does require your senses and reasoning. That is why I feel there should be a prerequisite class on logic and critical thinking prior to taking any math class pre-algebra and above.  After taking the prerequisites, logic and critical thinking should be merged into the rest of the mathematics instruction.  Additionally, the online resources such as Khan Academy and others should be used to supplement the limited time teachers have for instruction in classes. A certain amount of repetition and memorization is needed in mathematics, but neither approach should be the "only" way to educate.

Magic Squares: The Math that Drives Them

   

                   What is the Math behind Magic Squares?   

    A magic square is basically just an arithmetic sequence arranged in a special way.  What do we know about an arithmetic sequence?  An arithmetic sequence is a sequence of numbers each with a common difference. Looking at it a different way, it is a sequence of number by which each number in the sequence is found by adding a specific number to the first number in the sequence. For Example:  The sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11....

 is a  sequence  that starts with 1 and each successive number in the  sequence  is found by adding 1 to the previous number in the  sequence .   Now let’s say we want to know the sum of the first 9 digits in this sequence. We could just add them all up one at a time, but that generally takes more time. And with larger sequences it would take a really long time. So we will set up a formula.

Looking at a arithmetic sequence algebraically we need to set up some variables. We will denote the starting value with, a, and the common difference with x and each term is n so algebraically an arithmetic sequence looks like this A, (A + X),( A + 2X), (A +3X), (A +4X),  ….. (A + Nx),

Where (A + nX) is the n^th   term in the sequence.  s_n is the sum of the ne terms

When we wish to know the sum of n terms of the sequence what we are saying algebraically is

s_n =           a         +   (a + x)          +...+ (a + 2x) + …+ (a + (n-2) x) + (a + (n-1) x)

And in order to reduce this to a simple formula we can work with we will take the above sequence and reverse it and add it to itself.

     S_n =           a         +   (a + x)          +...+ (a + 2x) + …+ (a + (n-2) x) + (a + (n-1) x)

+   S_n = (a + (n-1) x) + (a + (n-2) x)  +…+ (a + 2x) +… +    (a + x)      +              a

2 S_n =   (2a + (n-1) x) + (2a + (n-1) x) + …+(2a + (n-1) x) +…+(2a + (n-1) x) 

Since we are working with sequences of n terms grouping the last part is as simple as taking

(2a + (n-1) x), and multiplying it by n

2 S_n = n (2a + (n-1) x)

Then when we divide by 2 we have S_n = n (2a + (n-1) x)

                                                                           2

The resulting equation is close to what we will be using for magic squares.
The only difference is that with
S_n = n (2a + (n-1) x) is used with linear sequences that can be graphed out on a

               2

number line.   In order to modify this for working with magic squares we simply square the n Inside the parenthesis ()
  which changes our equation to S_n = n (2a + (n²-1) x)   We do this

                                                                 2                     

 because (n-1) is used to represent last term and in magic squares the number of terms is the dimension squared so the last term would be (n²-1).

Magic squares: basic construction method

       How to construct a 3x3 magic square

         In this section we will be constructing the basic magic square with the dimension 3x3 starting value 1 and common difference 1. If you do not know what a magic square is please refer to the Magic Squares: Introduction section. It is also important to note that this method works with all magic squares that have an odd
dimension. i.e. 5x5 7x7 9x9.......
                                                                                                                       

1. )  First we will begin with an empty 3x3 array.

2.)   Next we place the starting value 1 in the middle of the top row.
                                     

                                                                                                                   

3.) Now we move right one space and up one space, 

but as we can see highlighted in yellow this places the  2

outside the bounds of the 3x3 square.When this happens we 

simply bring  the 2 down to the bottom  square of the column it is positioned over.

4.) In this step we start with the 2 and once again we go right         

 one space and up one space. And once again this leaves us

out side the bounds of the 3x3 square so we place the 3 at the 

beginning of the row it is outside of. 

5.) Here we start with the 3 and once again move right 

one space and up one space, but this time it puts us in an

occupied space. when that happens we simply place the 4        

underneath the 3. 

6.)   In this step we start with the 4 and again move right one

 space and up one space. This space is within the bounds of the 

square and unoccupied so we can simply leave the 5 here.                   

7. ) Here we start with the 5 and go one space right and one

space up and as we can see we can once again simply just                    

place the 6 in this unoccupied space

8.) This time when we move right one space and up one 

space. we are not only outside the bounds of the square 

but we are also on a diagonal In case this happens we                             

place the 7 underneath the 6.

9.) Here we start with the 7 and move right one space

and up one space. Here we are again out side the bounds

of the square  so  we simply take the 8 and place it at the 

beginning of the row it is beside.

10.) now in the final step we only have one space left where we could go ahead and place the 9 there how ever I am still going to illustrate that the pattern still applies when there is only one space.  so we  start with the 8 move right one space and up one space. and again we are out side the bounds of the square, so we will move the 9 to the bottom of the column it is over.

Next in the series we will be looking deeper into the math behind these squares. 

Magic Squares: introduction

       

             What are magic Squares ?

          Magic squares are an arrangement of numbers in a grid of various dimensions to where each row, column, and diagonal will sum up to the same number. The grid they are placed in is generally a N x N array split into N² cells.For example: a 3 x 3 square will have 9 cells meaning there will be 9 numbers in the square.

Here is an example of a 3 x 3 magic square.

     
       Top Row :                4 + 9 + 2 = 15
       Middle Row:             3 + 5 + 7 = 15
       Bottom Row:            8 + 1 + 6 = 15
       Left Column:             4 + 3 + 8 = 15
       Middle Column:        9 + 5 + 1 = 15
       Right column:            2 + 7 + 6 = 15
       First Diagonal            4 + 5 + 6 = 15
       Second Diagonal       2 + 5 + 8 = 15

       Magic squares are an exciting study of number arrangements. Mathematicians have been fascinated by magic squares for centuries.They can be both fun and educational. Magic squares have a deep relation to a branch of mathematics known as combinatorics. However,  most just use them for entertainment.                          

                A birthday trick for magic squares

All Magic squares have a few basic components that you can use to create one. First you need a starting value, that is the lowest number that will be in the square. Second you need to know the common difference between the numbers in the square.  For example in the above square it contains all of the numbers 1, 2, 3 , 4 , 5 , 6, 7 ,8 , 9  each number in the series is obtained by adding 1 to the previous.  so the common difference between terms is 1   The third component of a magic square is the dimension. For Example. The above square has 3 rows and 3 columns so its dimension is 3 x 3. 

     Now for our birthday trick we will restrict the dimension to a 3 x 3. You may also want to keep a copy of the above square so you can see the order you put the numbers into, or you can save it and print it off to use in the trick..
(there is a method for constructing magic squares without using an already created one as a reference but ill include that in my continuation of the series on magic squares)

1. Now what you will do is ask a person for the month and year they were born, or use your own birthday.You use the last 2 digits of the year they were born for simplicity(or if you are really comfortable with your mental math skills you could use the whole year)  
2. You take those 2 digits and use it as the starting value  and you use the numerical value for the month as the common difference. 
3.  And in your head you multiply the month times 4 and add it to the 2 digit year
4.  you write down the result where they cant see you writing 
5. then  you then put the piece of paper with the number on it in your pocket 
6. you then use the year and month of their birth to construct a magic square.
7. once constructed you show them that every row column and diagonal all have the same sum
8. next you have them divide that sum by 3 and tell you the answer 
9. you then pull the piece of paper out of your pocket and "magically" it is the same number

you can omit step 8 if you are comfortable multiplying  the number you wrote down in step 4 by 3 in your head

Ill show an example of this using Benjamin Franklin's birth day 01/17/1706

1.  01/17/1706
2.   .starting value =06 common difference=1
3. 4 x the month 1 + 06 = 4 + 6 = 10   (here i will multiply times 3 so i can omit step 8 since) 10 x 3 = 30
4. 30 ( not actually writing down  because this is an example)
5. also skipping this step because its an example
6. 
   

   

 7.    Top Row :                 9 + 14 + 7 = 30

       Middle Row:            8 + 10 + 12 = 30

       Bottom Row:             13 +6 + 11 =30

       Left Column:                9 +8 + 13 =30

       Middle Column:        14 + 10 + 6 =30

       Right column:           7 + 12 + 11 = 30

       First Diagonal           9+ 10 + 11 = 30

       Second Diagonal      7 +10 + 11 = 30

As you can see the magic constant is the same number we got in step 3 

Here is a side by side comparison of the square from the example and the basic square I placed at the top so you can see how you can use the basic square to construct a square for the trick. 

 

The next part in the series will be how to construct a magic square given any starting value,common difference, and dimension without using the basic square as a template. I will be posting this in about 12 hours so be sure to check back with us.

A brief history of the affects of Technology on Mathematics Collaboration

Mathematics and Technology

In the world we live in today, technology is constantly changing the way we do things. It has evolved the way we are able to teach and to learn.

In past centuries, collaboration with other professionals in your field was limited to carrying stone tablets, and sending hand written letters by way of messengers. I don’t even want to imagine discussing complex proofs by way of messenger pigeon.  

Beginnings of modern collaboration

The Plimpton Library tablet from around 1700 BCE, written in cuneiform, contains the most influential and profound mathematical insights that are still used today. It was written on wet clay, essentially by making impressions using wedged instruments. 
As a mathematician, this made collaboration with others a long process. You would have to mix your clay, engrave your theories , and if you did not make any typographical errors allow your clay to harden  .After your clay had hardened, you would then have to carry a big clay tablet with you across the dessert to collaborate with mathematicians in other towns.

However, this beat the alternatives. If they had simply written their theories in the sand and hoped it stayed until someone came to look at them, the famous Pythagorean Theorem, and Euclidian geometry might not have existed today.

As a result of the ineffective means of sharing knowledge, one can only wonder what great mathematical discoveries were completely lost to us. For all we know the tablet with the method for finding all of the nth primes with one simple equation is buried in the sand, or was shattered by someone falling while carrying it.

The Renaissance Era

Now skip ahead to the 1400’s. The hundred year’s war is over. The Byzantine Empire has fallen, and most importantly the invention of the printing press. Now mathematicians could share their work with a much wider audience. With all of the new innovations of this era, mathematicians still had the time lapse of sending their proofs by way of messenger. Collaboration was still more efficient, and produced more mathematical advancements than previous centuries. Number theory, calculus, and Newtonian physics all came from this era, along with many more.

The computer age

The computer age gives us a whole new meaning of collaboration, innovation, and the ability to calculate very large numbers quickly. Mathematicians all over the world began to use these amazing new machines to assist in making new discoveries.

Branching from this invention, come the intranet/internet allowing collaboration on a scale never before seen. No longer were we bound by waiting for the mail to deliver our work to other mathematicians to check our proofs. We could digitize and zap our theorems to all parts of the globe in minutes.

Social Media

As internet speeds improved, we were able to communicate in ways we never thought possible.  Community websites allow us to find and connect with others in our field of interest globally. We no longer are bound with traveling the globe to find a particular expert who is working on a similar project. Now we can just “tweet” our way into new partnerships, learning teams and even employment. 

Math Poetry: The pi poem

        This poem is a free verse poem I wrote to remember the digits in pi. How it works is it uses each digit in pi as the number of letters in each word for instance the first few words are " Now I view a world"
'Now' has 3 letters ' I ' is 1 letter ' View ' is 4 letters ' a ' is 1 letter and ' world ' is 5 letters.  or 3.1415

                 Now I view a world            

                 Imaginary in itself                              

                 Solid for those unstable believers                  

              Naively compliant 

                   Set in the ignorant seal

                        What can the thinkers say 

                      to clarify intricate codes

              is realness realness. 

                live a quizzical reality 

                                                            3.141592653589323844338326952884197
                                                                 

A sad story about Math

 

         Your change will be $3.63 mam...

        My wife worked as a cashier at Mcdonalds, and one day a lady came in and placed her order. Her total came up to $16.37. The lady paid with a $20 , and when my wife handed her back $3.63 the lady got furious.
She swore that she was owed $4.00 because her total was $16.37. My wife, being an excellent judge of character, could tell that the lady was being totally serious. She truly thought that her change was supposed to be $4.00. My wife tried to explain to her what her change was every way she could. The manager and other employees tried to explain it to her, and then finally another customer waiting behind her had to break out his cellphone calculator to show the lady what her change should have been. After looking at the number on the calculator, she apologized and grabbed her food and left extremely embarrassed.

                       The sad truth

       Unfortunately, this was not the first time this has happened. The number of people who are unable to even properly calculate change is staggering. It was an occurrence that happened a couple of times each week. Levels of math literacy are steadily declining, and it should not be ignored. There are many programs and alternate learning centers that offering after school and even weekend classes to those struggling with math. However, they are not everywhere.Small towns like the one I am from do not have those resources at hand, and any of the alternatives are more expensive than the average household can afford.  

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Converting a number from base 10 to base 6

steps to convert to base 6 from base 10


Ill illustrate by example and explain each step as I go.


First you will set up your problem.  lets say we want to convert 200 to base 6

the problem will be set up as follows  200 =      x  6  +        in the first  blank,     , we divide 200 by 6 and drop the remainder which is 33  which makes our problem look like this 200 = 33 x 6 +    
now for the second blank we place the remainder,  which we dropped in the first step  which gives us this:
 200 = 33 x 6 + 2

The next step is you take the 33 and repeat what you did in the first step and placing it under the first part of or problem   like this        

200 = 33 x 6 + 2
33   =  5  x 6 +3


Then you take the 5 and do the same thing. and you end up with  200 = 33 x 6 + 2
                                                                                                   33   =  5  x 6 +3
                                                                                                    5  =   0 x 6  + 5
You then will take the remainders and from bottom to top and you get 532


This method works when converting to other bases as well


                                                       

Many ways to multiply: Russian Multiplication

           Russian Multiplication 

Here we are going to discuss Russian multiplication. This method is also known as binary multiplication. The simplicity of this type of multiplication stems from the fact that you are only halving and doubling numbers then adding. I will demonstrate by example.       

                                                             Multiply 17 X 18 = ?

1.)                  17 | 18                         first we place the two numbers side by side and separate them into 2                                          

                                                          columns

                                                                                                                                                                           

2.)                   17 | 18                        in this step we started with the column on the right and cut 18 in half 

                            |  9                         to get 9 then we cut the 9 in half and dropped the remainder to get 4

                            |  4                         then we cut the 4 in half to get 2 and the 2 was cut to get 1. we stopped

                            |  2                         here since we got to one this is as far as we need to go for this method

                            |  1                          to work

                                                                                                                                                                              

3.)                  17 | 18                         Here we took the left column and instead of halving we doubled the

                      34 |  9                          17 and got 34 and doubled the 34 and got 68 doubled the 68 and got

                      68 |  4                            136 and doubled the 136 to get 272. and we stopped here because

                    136 |  2                              the 272 is in the same row as the 1

                    272 |  1

                                                                                                                                                                       

4.)                  17 | 18                      

                      34 |  9                          What we did here was we looked in the right column and found all of the 

                      68 |  4                            even numbers that are in the right column and crossed out the 

                    136 |  2                             entire row that had an even number in the right column 

                    272 |  1

                                                                                                                                                                       

  5.)                17 | 18                                 

                      34 |  9            1             in this final step we took the numbers that were not crossed out in the

                      68 |  4            272           left column and added them together, and as you see we have the 

                    136 |  2            +34                so 17 X 18 =  306

                    272 |  1            306

Once again I am going to explain that although I use the words, " easy, quick, and simple" , everyone should pick the types and methods that work best for them. I feel it is important to understand that there are multiple ways of doing things in mathematics and when we stop searching for new approaches in math we stop the heart of mathematics and destroy what its was originally created to do. Which is explore, explain, examine, and expand our understanding of the world around us.