Reflections about the maths daily in the world around us.
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Math education and apps.
Primes as the basic building blocks of numbers
Showing posts with label modular arithmetic. Show all posts
Showing posts with label modular arithmetic. Show all posts
Dancing numbers: Numbers Circus: Modulo choreography. ( And: Circus for to tame numbers, too ) After the post dedicated to dancing sort algorithms, I think it's time to show a program made and published in 2004, which it could be useful in teaching elementary mathematics.
Is the "Dancing numbers" Originally made for MathCats.com , it never has been published. Finally I posted it on my page so do not be missed for the moment. The program allows you to manipulate a lot of integers (up to 2000) with some simple rules: decide the number of columns that are to line up neatly from lowest to highest. (Also possible to arrange smaller amounts of numbers)
The program allows for reflection on the divisibility of numbers and remainders: It is also advisable to use the mode 9 columns, that shows the rule of divisibility for 9. (The sum of all members of any number equals the heading number of the column) And all the rules explained in https://en.wikipedia.org/wiki/Modular_arithmetic (related to addition, subtraction, multiplication and division (sometimes), and congruence). Here is a sample of how the program works:
In
February 2010, the magazine "Scientific American" ISSN 0210136X number
401 in the Mathematics Games section, published the article of Agustin Rayo
(philosophy professor at MIT) on: "bricks, locks and progressions." http://www.investigacionyciencia.es/files/3486.pdfThere are specific items on the RSA encryption method. And on the Theorem of Ben Green and Terry Tao The article touched me.And
what attracted me was the beautiful graphics representing natural
numbers in the form of combinations of spheres and worms were colored
graphic representation of primes. (See Scientific American article in *.pdf cited above).At nummolt.com, I spent many years trying to graph the numbers. You can see my work from 1997 that I mean: http://www.nummolt.com/nummolt/numdown.htm It is an old tool, based on the reflections made from quotations by Barbara
Scott Nelson and others, read online about children's learning when to
add and subtract with borrowing. But
while it is easy to graphically represent the addition and subtraction
of natural numbers is however much more difficult to make a simple
representation of the multiplication, the amount of calculations
involving multiplication algorithm that are used and taught in schools, after memorizing the "multiplication tables".There have been many attempts to make graphic representations of
multiplication multiplication Mayan eg: Mayan Multiplication video. And myself, I also made my attempts in my programs.But the mere fact that prime numbers as the basic bricks which are built from all natural numbers, had not ever considered. When I read the article, I sent a letter to Agustin Rayo to
tell him that I had really liked the article, and I naively I asked ifthe graphscorrespondedtoanyinvestigationbeing carriedout. He politely replied that the drawings he had done in the best way that
he had come to illustrate the article, and do not corresponded to any
investigation. At this point, I already had developed for years with Wendy Petti
program for Mathcats "Place Value Party":
http://www.mathcats.com/explore/age/placevalueparty.htmlIn this program we tried to show the value of the position of the numbers from birthday cakes with candles.A few years after, Ulrich Kortenkamp published the Place Value Chart: Seeing Ulrich Kortenkamp program I wrote to MathForum saying that this
program was a lesson for me, and I congratulate the author:http://mathforum.org/mathtools/tool/181488/ Few years later,talking with JoanJareñoabouta game withprimes,fromCreamattheywarnedme about the existence ofaposterwithprimes: http://esquemat.es/algebra/factores-primos-por-colores In reference to the original from John Graham-Cumming:http://blog.jgc.org/2012/04/make-your-own-prime-factorization.html At this time (2013), I understood that I had all the pieces to build a tool that I dreamed. When I was looking at the internet to address program development, in addition to the well-known Ulam spiral, I had knowledge of the parabolic sieve: shown by Yuri Matiyasevich and Boris Stechkin form the Steklov Mathematical Institute of the Russian Academy of Sciences. There's an explanation for this, here: http://plus.maths.org/content/catching-primes
The first had to do was to assign a color to each prime number.Because the app is dedicated to the children, I decided to give the
three basic colors first three prime numbers, and intersperse the
following numbers.Therefore, red is 2, green 3 and 5 is blue, 7 yellow, magenta 11 cyan and 13. And from there, putting the colors go.The purpose of this distribution is that the resulting color of the
product color is the sum of the colors of the prime factors or filtration of colors of the prime numbers.Thus, the color corresponding to 30 (2 * 3 * 5) will be white, and the color corresponding to 1001 (7 * 11 * 13) will be black. Having decided this, the work was left to do was clear:Show prime numbers as small circles within a larger circle, usually corresponding to a composite number.Removing prime circles from the circumference, equals to divide.Add a prime circle, equivalent to multiply.In parallel, the the app displays numbers in a place value format, but in vertical, as in the Mathcats "Place Value
Party" program. And adding the display parabolic sieve, and all the numbers well placed within the Ulam spiral and the representation of the module number. In the end, the program Touch Natural Numbers is a small laboratory that can be studied composition and numbers, and make elementary operations within the set of natural numbers. And never there is a division that is not resulting integer. Nor has there ever the possibility of subtraction with a negative result.
Hope you like it, and especially useful for teaching math in elementary school.
