Touch Fraction ℚ (1.4.x)

New version of Android App "Touch Fraction ℚ"  (version 1.4.x and up)
https://play.google.com/store/apps/details?id=nummolt.touch.fraction


Evolution of the app over the years: (1993-2015)
In 1993 I developed the first executable (for Windows 3.1) of Touch Fraction ℚ (Racional.exe):

http://www.nummolt.com/rebost/index.html

It was my first attempt of to show an interactive representation of Rational numbers as fibres in (Z * Z {0}) and arrangement in Q.
As the Rational representation in Wikipedia:  Rational Numbers:

 https://en.wikipedia.org/wiki/Rational_number

In 2013 I made the new version of the app: The translation of the original executable to the Android OS.

The interaction became extended to the fraction:

The pie fraction to navigate across the fractions, and the rational representation to navigate across the equivalent fractions. (and select the range of available denominators)

In 2015 after the development of Touch Integers ℤ (+ - × ÷) (https://play.google.com/store/apps/details?id=com.nummolt.touch.integers)

This time, I was able to afford the next step:

Explain fractions and rational numbers as the prime factorization of its members: numerator and denominator.

This is the new version of Touch Fraction ℚ:

https://play.google.com/store/apps/details?id=nummolt.touch.fraction

Interact with the fraction, build fractions adding or removing prime factors in the numerator or in the denominator.

Simplify fractions dragging common prime fractions to the "common" zone, and invert fractions with the "^-1" button.  

Touch Fraction ℚ is a complete tool to understand positive fractions, negative fractions, positive and negative numerators, positive and negative denominators, equivalent fractions, and inverted fraction.

(Video of the app, at: Google Play)

Ancient garden machinery

Some days ago, I stumbled upon this page:

Gardens as crypto water computers

http://pruned.blogspot.com.es/2012/01/gardens-as-crypto-water-computers.html

There, there's also a reference to the work of Chandra Mukerji:  

"Territorial ambitions and the gardens of Versailles".

https://books.google.es/books?id=TpJvLySgrDwC

Chandra Mukerji - Cambridge University Press, Sep 25, 1997 - History - 393 pages

There: "Chandra Mukerji highlights the connections between the seemingly disparate activities of engineering and garden design, showing how the gardens at Versailles showcased French skills in using nature and art to design a distinctively French landscape and create a naturalized political territoriality."

More territorial models:

Mississippi River Basin Model:

https://en.wikipedia.org/wiki/Mississippi_River_Basin_Model

 

 

 

 

 

From my point of view is a suggestive research field. I created a group google plus to continue the search :

"Ancient garden machinery"  (broken link, obvously)

https://plus.google.com/communities/112320067568116153542

If someone wants to help, he will be well received in the group.

Touch Integers ℤ (+ - × ÷)

Touch Integers is the evolution of the Touch decimals Place value ±. (in the same blog)
Touch decimals could not easily multiply or divide numbers:

I've started my reflections about this 20 years ago:

Is very easy add and subtract graphically. One can regroup the tokens of each order, regroup, carry or borrow tokens, and you can obtain the result in a simulation of abacus.

But not so easy to practice multiplication or division in this visual and interactive way

I looked the inside of the numbers:

Inside the numbers there are the components of the number: The prime factors.

To multiply two integers you must regroup the components of the two numbers.

To divide a integer, you must separate the components.

The program only works with integers. adds, subtract, multiplies and divides (but only exact division) 

Is my latest Android App: 

At Google Play: 
https://play.google.com/store/apps/details?id=com.nummolt.touch.integers
I hope you will find it useful for teaching.

Some animations:

At left: two abacuses (two numbers stacked). 
At right two circles with the prime factors. (two circles with prime numbers stacked)
At right edge: all the prime numbers available to the app. 

To create a number:  Tap on the cells at left. The app shows the number
To add: Drag the tokens from one abacus to the other.
To subtract: Tap the sign key and drag from one abacus to the other.
To multiply: (the numbers must be previously created with the earlier previous steps)
Drag from one prime circle to the other prime circle.
To Divide a number:
Drag the prime numbers outside the prime circle:
Release prime factors to the other prime circle (integer division and multiplication) 
Release prime factors between the prime circles (integer division)
Release prime factors in the list of the right edge: (integer division and erase the prime factor)
Scroll and pick a prime number from the list of the right edge:

And release it in the free zone, or in a prime circle (multiplication) 

Playing with 12*12:

Creating two prime numbers: 
Multiply them. 
Restoring to the original state.
Throwing the prime numbers to the primes list. 

Picking numbers from the big list of prime numbers:
Playing with 2; 3; 5; 7; 11; 13.
1001; 30 and 30,030

(In the current version the top prime number available is 19,874,419)

Multiplication Table with Prime Numbers made with elements from Touch Integers:

 

 

///////////////////////////////////////////////////ACKNOWLEDGEMENTS///////////////////////////////////////////////////////

(Expanded from the Google Play page https://play.google.com/store/apps/details?id=com.nummolt.touch.integers)

Jacobo Bulaewsky: (Arcytech.com (broken)) (12/08/1955 - 08/25/2004)

http://mathforum.org/mathtools/search/index.html?execute=search&database=catalog&keywords=Bulaevsky

There's a software restoration made by Jill Britton.

Brian Sutherland: ( http://www.our-montessori.ca ) Montessori methods adapted to computer. 

(Shockwave Player activities: covering addition, subtraction, multiplication and division) 
Long multiplication: https://www.youtube.com/watch?v=qDMXNjtuqqo
Long division: https://www.youtube.com/watch?v=seJC-1BR_gQ 

Agustín Rayo: (Philosopy professor at MIT) 

And his article about the Prime Numbers, at Scientific American (Spanish version - 02/2010)):

http://www.investigacionyciencia.es/files/3486

"Ladrillos, candados y progresiones.

El fabuooso mundo de los números primos".

Agustín Rayo (2010): http://www.investigacionyciencia.es/files/3486

Ulrich Kortenkamp: (Professor für Didaktik der Mathematik. Universität Potsdam. 

Author of "Place Value Chart" and "Cinderella") http://kortenkamps.net/index.php/Hauptseite 

Place Value Chart: Web page: Stellenwerttafel: http://kortenkamps.net/index.php/Stellenwerttafel

Place Value Chart: (iOS) https://itunes.apple.com/de/app/stellenwerttafel/id568750442 

Cinderella: http://cinderella.de/tiki-index.php

Ulrich Kortenkamp: Place Value Chart

Christian Urff: http://www.lernsoftware-mathematik.de/?p=1331

Math Tablet: https://play.google.com/store/apps/details?id=air.com.urff.test.TeileGanzes

Wendy Petti (Teacher and author of MathCats): http://www.mathcats.com 

Nearly 20 years of support and patience with me. And specially for the vision about the position of the places in "Touch decimals Place value ±" which allowed continuing the work well.

Our first work as a team: OBBL Architecture blocks: http://www.mathcats.com/explore/obbl/obbl.html
And "Place Value Party Cake":

MathCats: "Place value party": http://www.mathcats.com/explore/age/placevalueparty.html

Jeff LeMieux: (Builder, teacher along 35 years and software developer) 
Scripts Web Page: http://syzygy.virtualave.net/

For his work and the assistance in the development of Touch Decimals: Option without negative numbers.

Joan Jareño (From: CREAMAT team) http://srvcnpbs.xtec.cat/creamat/joomla/ 

Author of "Calaix +ie": http://xtec.cat/~jjareno/    http://calaix2.blogspot.com.es/

And History of numbers: Calculus: http://xtec.cat/~jjareno/calculus/

For their help in the last steps in the development of "Touch Integers".
* * *

Added later:
Playing with the app "Touch Integers ℤ (+ - × ÷)":
Exploring Mersenne primes: 2^p -1 (Not all are primes) 
nummolt.blogspot.com/2016/06/exploring-mersenne-primes-2p-1.html
https://www.youtube.com/watch?v=sOnkpDihIS4



  

Dancing Sort Algorithms

Post, only to illustrate the most common types of sorting algorithms:
Graphically, Dance, and Code:

(Graphic from http://www.sorting-algorithms.com/ )
(Videos from: https://www.youtube.com/user/AlgoRythmics )
(Pseudocode: http://visualgo.net/ )
(Pseudocode Shell: https://courses.cs.washington.edu/courses/cse326/03wi/lectures/RaoLect14.pdf )
(Main Links (titles): Wikipedia: https://en.wikipedia.org/wiki/Sorting_algorithm ) 

Sorting Algorithms general animated image:
(Algorithms race)

( from: http://www.sorting-algorithms.com/ )

1.-  Insert:

mark first element as sorted

for each unsorted element

  'extract' the element

  for i = lastSortedIndex to 0

    if currentSortedElement > extractedElement

      move sorted element to the right by 1

    else: insert extracted element

2.- Select:

repeat (numOfElements - 1) times

  set the first unsorted element as the minimum

  for each of the unsorted elements

    if element < currentMinimum

      set element as new minimum

  swap minimum with first unsorted position

3.- Bubble:

do

  swapped = false

  for i = 1 to indexOfLastUnsortedElement

    if leftElement > rightElement

      swap(leftElement, rightElement)

      swapped = true

while swapped

4.- Shell:

                                # Start with the largest gap and work down to a gap of 1

                                     foreach (gap in gaps){

                                # Do a gapped insertion sort for this gap size.
                                                     # The first gap elements a[0..gap-1] are already in gapped order

                                                     # keep adding one more element until the entire array is gap sorted  

                                     for (i = gap; i < n; i += 1){

                                            temp = a[i]

                                            for (j = i; j >= gap and a[j - gap] > temp; j -= gap){

                                                  a[j] = a[j - gap]

                                            }

                                            a[j] = temp

                                      }

                                }

5.- Merge:

split each element into partitions of size 1

recursively merge adjancent partitions

  for i = leftPartStartIndex to rightPartLastIndex inclusive

    if leftPartHeadValue <= rightPartHeadValue

      copy leftPartHeadValue

    else: copy rightPartHeadValue

copy elements back to original array

7.- Quick:

for each (unsorted) partition

  set first element as pivot

  storeIndex = pivotIndex + 1

  for i = pivotIndex + 1 to rightmostIndex

    if element[i] < element[pivot]

      swap(i, storeIndex); storeIndex++

  swap(pivot, storeIndex - 1)

Touch decimals. Part 5. Collaboration

The 07/04/2015 after some days from the first post of "Touch decimals":
https://play.google.com/store/apps/details?id=com.nummolt.math.value.place 
I received this animation from JeffL (member of MathTools & author of: 
"Jeff's Interactive K12 Math Scripts Page") :

http://www.nummolt.com/LeMieux.gif 

This led me to develop an alternative program with its version.
And I added it to my original program.
With translation between the two systems.
(One with 'Tokens' signed and  'Places' unsigned, and the other with color
'Tokens'  unsigned and 'Places' signed).
I think the program has been enriched by the two visions.

The pre-release videos:

Borrowing in subtraction

Borrowing in subtraction and the color of Tokens

Transition between the two methods

The YouTube Video about the posted version (July 15 2015): Touch decimals v.1.1.0 r.4:

Touch decimals promotional video

I should also like to thank Wendy Petti for comments made in the early development of this program.

From 2015/09/1 There's a online version of "Touch decimals Place value"  JavaScript version:

www.nummolt.com/touchdecimals/index.html 

Touch decimals: Part 4: Borrowing in subtraction. Carry in Addition.

Touch decimals Place value ±

Borrowing in subtraction:

Carry in addition:

MERGED 2016/11/10:

And from a dialog on Twitter: https://twitter.com/macajo/status/793790918788837376

The following Video:  Subtraction: 534-486:

 

Touch decimals Part 3: Basic interaction

Touch decimals,Place value ±

Basic interactions:

Video: